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Hey everyone,
As I mentioned in my response about my solve of the 120Cell, I have been
looking into some other 4D puzzles and have worked out how several of these
puzzles should work and even discovered some interesting properties. Here is
a snipet from my 120Cell solve message I sent Roice discussing this subject:
"I am still hoping for more complicated 4D puzzles and am willing to do
whatever I can to help make them a reality. Coding a 4d space like you have
is quite intimidating, but perhaps I can try to build off a pre-existing one
with some guidance. I have already determined what the 4D analogue of the
FTO (face turning octahedron, invented some time last year if you have not
already seen it) would look like and how it would function as well as the 4D
analogue of the Skewb and Helicopter Cube (on that note I also have a
suggestion as to how to make the interface for 4D puzzles that are non-face
rotating, like the Skewb and Helicopter Cube). I have also made some
interesting discoveries like for example making a 4D puzzle out of a 3D
puzzle can make some additional internal cuts without altering the exterior
of a 3D face (true for all three puzzle I mentioned so far) and how a 4D
Skewb is not deepcut! (that is every cell looks like a Skewb and seems to
behave as such) The vertex turning deepcut hypercube has faces that
externally each look like a dino cube. Is there anything I can do to make
help make these a reality? After spending 150 hours on the 120Cell, I can
honestly say that about 146 of the hours all feel exactly the same and I am
dying to find a more interesting 4D puzzle to explore :)"
To expand a little on some of the things I mentioned above, the 4D FTO would
be a 24Cell with faces that look like an exploded version of this puzzle:
http://www.jaapsch.net/puzzles/octaface.htm
with one big difference, in addition to every cut on the 3D analogue of the
puzzle, the 4D version has and additional cut perpendicular to the vertices
of each face that line up with first cut down. :/ Sorry, I know that wasn't
very well worded and I'm not sure how well sending a picture would work
through a yahoo group. Let me try again: these extra cuts would essential
cut off the vertex pieces of each cell. Removing the pieces that are
affected by this new unexpected cut will result in cells that have an
exterior that matches this puzzle:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=451
(If you can follow my inadequate descriptions above, the 4D FTO would have 6
distinct visible pieces, not just the 5 present on an exploded 3D FTO - the
extra comes from splitting each of the vertex pieces of the 3D Fto in half)
A similar phenomenon occurs on both the 4D helicopter cube (3D:
http://www.puzzleforge.com/main/index.php?option=com_content&view=article&id=49:hcannounce&catid=1:latest-news&Itemid=50)
and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by analogue, I
mean each cell looks like the respective puzzle and moves in a similar
manner]. In each of these puzzles, the new cut clips off the corners.
Remembering that to truly express the 4D nature of these puzzles, each cell
must be "exploded", so what used to be he vertex pieces for each of these
puzzles have now been cut in half resulting in an internal piece that
behaves as one might have expected the single original piece to act and an
external piece that in addition to moving every time the internal piece
moves, can also be affected by a non-adjacent face.
As to a nice interface for non-face rotating 4D puzzles, my suggestion is to
display the wireframe of a 3D solid that displays all the symmetries implied
by the rotation between the faces and perform clicks not on the puzzle
itself, but only on this wireframe. For example, on a 4D Skewb, rotations
are made around the "corners" of each cell. These rotations are all
equivalent to some rotation on a face turning 16Cell. So, in the Hypercube
shape, we could display wireframes of tetrahedrons that "float" between the
appropriate corners of 4 hypercube cells. When the user clicks on a face of
this floating wirefram tetrahedron, both the tetrahedron and the pieces
affected by the corresponding "vertex twist" all rotate. Clicking on the
actual stickers of the puzzle does nothing; all rotations are executed by
clicking on these "rotation polyhedra". In the case of the 4D Helicopter
Cube, the appropriate wireframe shape would be a triangular prism -
rotations around both the triangle faces and the rectangular faces are
possible moves on the 4D Helicopter Cube, and each of these rotations can be
executed unambiguously by clicking on the appropriate face of the triangular
prism wireframe floating between the cells of the puzzle.
As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, that is -
a hypercube consisting of exploded skewbs (with additional cuts clipping off
the corners). Now identify all the pieces affected by one particular
rotation and try to identify the move that is on the opposite side of the
puzzle. Identified correctly, this opposite move does not affect any of the
same pieces. However, not every piece is affected by these two moves! There
is a band of pieces remaining untouched, much like the slice of a 3x3x3 left
untouched by UD'. This means the puzzle is not deepcut! If we push the 3D
hyper cutting planes deeper into the 4D puzzle, we get cells that look like
Master Skewbs. Continuing to push, certain pieces of these Master Skewbs get
thinner and thinner until they vanish at the point when opposing hyperplanes
meet. This is the deepcut vertex turning 8Cell puzzle. Each cell looks like
an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells that
look like dino cubes that is shallower cut. Although these puzzles are
visually identical, a single move on the shallower cut puzzle affects pieces
on only 4 cells while a single move on the deepcut puzzle affects pieces on
all 8 cells. Also of interest is the series of complicated looking puzzles
that appear at cut depths between the 4D Skewb and each of these dino cell
puzzles, although there are only 3 slices per axis in these puzzles (same
order as 3x3x3), each cell is an exploded Master Skewb!
Although I have explored several other ideas, the three puzzles (4D FTO, 4D
Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
candidates for the next run of 4D puzzles, they implement some complex piece
interactions without becoming too large or too visually crowded.
These puzzles are of an incredible interest to me, because the interactions
of the pieces are so much more intricate than the 120Cell or any of the
simplex vertex puzzles possible in the current MC4D program! As I mentioned
in my message to Roice, I have a good idea of how each of these puzzles look
and function and would gladly assist anyone (Roice? haha) who wants to
attempt to program it. In the meantime, I will take a look at the code Roice
has provided me and try to do some work myself, but I highly doubt I will
have success without an experienced programmer's help ;)
I would love to hear others' thoughts on these!
-Matt Galla
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Hi Matt,
Thank you for starting the discussions about other 4D puzzles.=20
Can you explain more about why the 4D analogue of the FTO is a 24-cell inst=
ead of a 16-cell? Although the faces of the 24-cell are octahedra, 24-cell =
is a self-dual polytope that is not a simplex. From this point of view, it =
has no 3D analog. In fact it has no analog in any dimension other than 4D. =
However, the 16-cell belongs to the family of cross-polytopes, which are th=
e duals of hypercubes, and exist in any number of dimensions. (http://en.wi=
kipedia.org/wiki/Cross-polytope). In 3D, the cross-polytope is 16-cell. The=
refore I think a natural extension of FTO is a cell-turning 16-cell, becaus=
e they share more similarities.
For example, you may know that in 3D, the FTO can be regarded as a shape-mo=
d of Rex Cube, a vertex turning cube (http://www.twistypuzzles.com/forum/vi=
ewtopic.php?f=3D15&t=3D12659). If the 4D FTO is a shape-mod of the vertex t=
urning hypercube, it should be a cell-turning 16-cell instead of a cell-tur=
ning 24-cell.=20
No matter calling it 4D FTO or else, I believe what you have described in t=
he third paragraph is a cell-turning 24-cell. It should be an amazing puzzl=
e to solve. I have special feeling about it because of its uniqueness in al=
l the dimensions.=20
Nan
--- In 4D_Cubing@yahoogroups.com, "Galla, Matthew"
>
> Hey everyone,
>=20
> As I mentioned in my response about my solve of the 120Cell, I have been
> looking into some other 4D puzzles and have worked out how several of the=
se
> puzzles should work and even discovered some interesting properties. Here=
is
> a snipet from my 120Cell solve message I sent Roice discussing this subje=
ct:
>=20
> "I am still hoping for more complicated 4D puzzles and am willing to do
> whatever I can to help make them a reality. Coding a 4d space like you ha=
ve
> is quite intimidating, but perhaps I can try to build off a pre-existing =
one
> with some guidance. I have already determined what the 4D analogue of the
> FTO (face turning octahedron, invented some time last year if you have no=
t
> already seen it) would look like and how it would function as well as the=
4D
> analogue of the Skewb and Helicopter Cube (on that note I also have a
> suggestion as to how to make the interface for 4D puzzles that are non-fa=
ce
> rotating, like the Skewb and Helicopter Cube). I have also made some
> interesting discoveries like for example making a 4D puzzle out of a 3D
> puzzle can make some additional internal cuts without altering the exteri=
or
> of a 3D face (true for all three puzzle I mentioned so far) and how a 4D
> Skewb is not deepcut! (that is every cell looks like a Skewb and seems to
> behave as such) The vertex turning deepcut hypercube has faces that
> externally each look like a dino cube. Is there anything I can do to make
> help make these a reality? After spending 150 hours on the 120Cell, I can
> honestly say that about 146 of the hours all feel exactly the same and I =
am
> dying to find a more interesting 4D puzzle to explore :)"
>=20
> To expand a little on some of the things I mentioned above, the 4D FTO wo=
uld
> be a 24Cell with faces that look like an exploded version of this puzzle:
> http://www.jaapsch.net/puzzles/octaface.htm
> with one big difference, in addition to every cut on the 3D analogue of t=
he
> puzzle, the 4D version has and additional cut perpendicular to the vertic=
es
> of each face that line up with first cut down. :/ Sorry, I know that wasn=
't
> very well worded and I'm not sure how well sending a picture would work
> through a yahoo group. Let me try again: these extra cuts would essential
> cut off the vertex pieces of each cell. Removing the pieces that are
> affected by this new unexpected cut will result in cells that have an
> exterior that matches this puzzle:
> http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=3D451
> (If you can follow my inadequate descriptions above, the 4D FTO would hav=
e 6
> distinct visible pieces, not just the 5 present on an exploded 3D FTO - t=
he
> extra comes from splitting each of the vertex pieces of the 3D Fto in hal=
f)
>=20
> A similar phenomenon occurs on both the 4D helicopter cube (3D:
> http://www.puzzleforge.com/main/index.php?option=3Dcom_content&view=3Dart=
icle&id=3D49:hcannounce&catid=3D1:latest-news&Itemid=3D50)
> and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by analogue,=
I
> mean each cell looks like the respective puzzle and moves in a similar
> manner]. In each of these puzzles, the new cut clips off the corners.
> Remembering that to truly express the 4D nature of these puzzles, each ce=
ll
> must be "exploded", so what used to be he vertex pieces for each of these
> puzzles have now been cut in half resulting in an internal piece that
> behaves as one might have expected the single original piece to act and a=
n
> external piece that in addition to moving every time the internal piece
> moves, can also be affected by a non-adjacent face.
>=20
>=20
> As to a nice interface for non-face rotating 4D puzzles, my suggestion is=
to
> display the wireframe of a 3D solid that displays all the symmetries impl=
ied
> by the rotation between the faces and perform clicks not on the puzzle
> itself, but only on this wireframe. For example, on a 4D Skewb, rotations
> are made around the "corners" of each cell. These rotations are all
> equivalent to some rotation on a face turning 16Cell. So, in the Hypercub=
e
> shape, we could display wireframes of tetrahedrons that "float" between t=
he
> appropriate corners of 4 hypercube cells. When the user clicks on a face =
of
> this floating wirefram tetrahedron, both the tetrahedron and the pieces
> affected by the corresponding "vertex twist" all rotate. Clicking on the
> actual stickers of the puzzle does nothing; all rotations are executed by
> clicking on these "rotation polyhedra". In the case of the 4D Helicopter
> Cube, the appropriate wireframe shape would be a triangular prism -
> rotations around both the triangle faces and the rectangular faces are
> possible moves on the 4D Helicopter Cube, and each of these rotations can=
be
> executed unambiguously by clicking on the appropriate face of the triangu=
lar
> prism wireframe floating between the cells of the puzzle.
>=20
>=20
> As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, that i=
s -
> a hypercube consisting of exploded skewbs (with additional cuts clipping =
off
> the corners). Now identify all the pieces affected by one particular
> rotation and try to identify the move that is on the opposite side of the
> puzzle. Identified correctly, this opposite move does not affect any of t=
he
> same pieces. However, not every piece is affected by these two moves! The=
re
> is a band of pieces remaining untouched, much like the slice of a 3x3x3 l=
eft
> untouched by UD'. This means the puzzle is not deepcut! If we push the 3D
> hyper cutting planes deeper into the 4D puzzle, we get cells that look li=
ke
> Master Skewbs. Continuing to push, certain pieces of these Master Skewbs =
get
> thinner and thinner until they vanish at the point when opposing hyperpla=
nes
> meet. This is the deepcut vertex turning 8Cell puzzle. Each cell looks li=
ke
> an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells tha=
t
> look like dino cubes that is shallower cut. Although these puzzles are
> visually identical, a single move on the shallower cut puzzle affects pie=
ces
> on only 4 cells while a single move on the deepcut puzzle affects pieces =
on
> all 8 cells. Also of interest is the series of complicated looking puzzle=
s
> that appear at cut depths between the 4D Skewb and each of these dino cel=
l
> puzzles, although there are only 3 slices per axis in these puzzles (same
> order as 3x3x3), each cell is an exploded Master Skewb!
>=20
> Although I have explored several other ideas, the three puzzles (4D FTO, =
4D
> Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
> candidates for the next run of 4D puzzles, they implement some complex pi=
ece
> interactions without becoming too large or too visually crowded.
>=20
> These puzzles are of an incredible interest to me, because the interactio=
ns
> of the pieces are so much more intricate than the 120Cell or any of the
> simplex vertex puzzles possible in the current MC4D program! As I mention=
ed
> in my message to Roice, I have a good idea of how each of these puzzles l=
ook
> and function and would gladly assist anyone (Roice? haha) who wants to
> attempt to program it. In the meantime, I will take a look at the code Ro=
ice
> has provided me and try to do some work myself, but I highly doubt I will
> have success without an experienced programmer's help ;)
>=20
> I would love to hear others' thoughts on these!
> -Matt Galla
>
Matt,
My plan after 5D simplex is some 4D puzzle. It may be 24cell, or hypercub=
e with some different cutting (1C-centered and 4C-centered? don't know). Fo=
r the twists I'm going to use free rotation of layer: you select it (face b=
y left-click, 2C/3C/4C if there are such centers - by right click) and then=
use free rotation in 3D around the fixed center. I also thought about addi=
tional stickers - but it's equivalent to work with some truncated puzzle: a=
dditional faces will play the role of stickers for non-face-centered layers=
.
But before there will be time to write description of log files for the w=
ide class of 4D puzzles. I think that it must include:
- symmetry of the body (set of possible rotation axes)
- number of layers for each class of axes (e.g. hypercube, 3 layers in 1C d=
irection, 2 layers in 4C direction)
- sequence of twists in terms "direction of rotation center, direction of r=
otation axis, angle, layer mask"
All the other things (actual shape of main body, coloring, depths of cuts) =
are not so important - they are about the puzzle description and if two pro=
grams will recognize, say, the same name in the similar way, they will give=
same interpretation of the rest of log file.
Andrey=20
Hi Matt, Hi Matt,
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Ah,
schuma (Nan?) is quite right. A very "natural" (perhaps even the most
"natural") extension of the FTO to 4D is a cell turning 16Cell. However, I
was looking for a puzzle where each cell looks like an FTO, and this
obviously cannot be the case for a 16Cell, which has tetrahedral faces.
It seems that in 4D there are two ways of interpreting the analogue of some
puzzles.
On the one hand, you could construct a 4D puzzle where every cell looks lik=
e
the 3D counterpart, in the case of all puzzles except tetrahedral and
icosahedral, this unambiguously assigns the 4D shape. In the case of a
tetrahedral puzzle, you can choose between the 5Cell, the 16Cell, and the
600Cell. In the case of icosahedral, this interpretation fails to produce a=
n
equivalent puzzle.
On the other hand, you can analyze the construction of the 3D shape and
construct the equivalent 4D shape. In the case of the octahedron, 4
triangles meet at a point (triangle being the 2-simplex). Thus the 4D
equivalent should have 4 tetrahedra (tetrahedron being the
3-simplex) meeting at an edge. This can unambiguously find analogues for al=
l
regular polyhedra (in the case of the FTO, this interpretation gives a
16Cell with pyraminx-like cells, the one schuma is referring to), and
possibly more; however no puzzle will ever get mapped to the 24Cell (becaus=
e
the 24Cell has no 3D equivalent).
I realize the first method given above is "artificial" in a sense. You do
not design a 3D puzzle by first deciding what each face should look like an=
d
then repeating it over the rest of the puzzle. BUT YOU COULD! ;) As long as
you pick a face that is cut in such a way that all cuts are parallel to the
sides of the face-shape and at equal depths, the resulting puzzle should be
"playable". (the 4D analogue for this is choosing a cell layout such that
all cuts are parallel to the faces of the cell and at equal depths - but
this is PRECISELY what allows the cell to alone be a 3D puzzle)
In any case, it seems that both methods produce valid puzzles, and while
some 4D puzzles can be obtained through either interpretation, there are
some (like the 24Cell 4D FTO I described earlier) that can only be produced
through one interpretation. I therefore think it is important that we
consider both interpretations (plus I think a 24Cell would be more exciting=
,
but maybe that's just me ;) )
Thanks for bringing that up schuma!
-Matt Galla
PS On TP my username is Allagem ;)
On Sun, Jan 23, 2011 at 12:44 PM, schuma
>
>
> Hi Matt,
>
> Thank you for starting the discussions about other 4D puzzles.
>
> Can you explain more about why the 4D analogue of the FTO is a 24-cell
> instead of a 16-cell? Although the faces of the 24-cell are octahedra,
> 24-cell is a self-dual polytope that is not a simplex. From this point of
> view, it has no 3D analog. In fact it has no analog in any dimension othe=
r
> than 4D. However, the 16-cell belongs to the family of cross-polytopes,
> which are the duals of hypercubes, and exist in any number of dimensions.=
(
> http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polytope i=
s
> 16-cell. Therefore I think a natural extension of FTO is a cell-turning
> 16-cell, because they share more similarities.
>
> For example, you may know that in 3D, the FTO can be regarded as a
> shape-mod of Rex Cube, a vertex turning cube (
> http://www.twistypuzzles.com/forum/viewtopic.php?f=3D15&t=3D12659). If th=
e 4D
> FTO is a shape-mod of the vertex turning hypercube, it should be a
> cell-turning 16-cell instead of a cell-turning 24-cell.
>
> No matter calling it 4D FTO or else, I believe what you have described in
> the third paragraph is a cell-turning 24-cell. It should be an amazing
> puzzle to solve. I have special feeling about it because of its uniquenes=
s
> in all the dimensions.
>
> Nan
>
>
> --- In 4D_Cubing@yahoogroups.com <4D_Cubing%40yahoogroups.com>, "Galla,
> Matthew"
> >
> > Hey everyone,
> >
> > As I mentioned in my response about my solve of the 120Cell, I have bee=
n
> > looking into some other 4D puzzles and have worked out how several of
> these
> > puzzles should work and even discovered some interesting properties. He=
re
> is
> > a snipet from my 120Cell solve message I sent Roice discussing this
> subject:
> >
> > "I am still hoping for more complicated 4D puzzles and am willing to do
> > whatever I can to help make them a reality. Coding a 4d space like you
> have
> > is quite intimidating, but perhaps I can try to build off a pre-existin=
g
> one
> > with some guidance. I have already determined what the 4D analogue of t=
he
> > FTO (face turning octahedron, invented some time last year if you have
> not
> > already seen it) would look like and how it would function as well as t=
he
> 4D
> > analogue of the Skewb and Helicopter Cube (on that note I also have a
> > suggestion as to how to make the interface for 4D puzzles that are
> non-face
> > rotating, like the Skewb and Helicopter Cube). I have also made some
> > interesting discoveries like for example making a 4D puzzle out of a 3D
> > puzzle can make some additional internal cuts without altering the
> exterior
> > of a 3D face (true for all three puzzle I mentioned so far) and how a 4=
D
> > Skewb is not deepcut! (that is every cell looks like a Skewb and seems =
to
> > behave as such) The vertex turning deepcut hypercube has faces that
> > externally each look like a dino cube. Is there anything I can do to ma=
ke
> > help make these a reality? After spending 150 hours on the 120Cell, I c=
an
> > honestly say that about 146 of the hours all feel exactly the same and =
I
> am
> > dying to find a more interesting 4D puzzle to explore :)"
> >
> > To expand a little on some of the things I mentioned above, the 4D FTO
> would
> > be a 24Cell with faces that look like an exploded version of this puzzl=
e:
> > http://www.jaapsch.net/puzzles/octaface.htm
> > with one big difference, in addition to every cut on the 3D analogue of
> the
> > puzzle, the 4D version has and additional cut perpendicular to the
> vertices
> > of each face that line up with first cut down. :/ Sorry, I know that
> wasn't
> > very well worded and I'm not sure how well sending a picture would work
> > through a yahoo group. Let me try again: these extra cuts would essenti=
al
> > cut off the vertex pieces of each cell. Removing the pieces that are
> > affected by this new unexpected cut will result in cells that have an
> > exterior that matches this puzzle:
> > http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=3D451
> > (If you can follow my inadequate descriptions above, the 4D FTO would
> have 6
> > distinct visible pieces, not just the 5 present on an exploded 3D FTO -
> the
> > extra comes from splitting each of the vertex pieces of the 3D Fto in
> half)
> >
> > A similar phenomenon occurs on both the 4D helicopter cube (3D:
> >
> http://www.puzzleforge.com/main/index.php?option=3Dcom_content&view=3Dart=
icle&id=3D49:hcannounce&catid=3D1:latest-news&Itemid=3D50
> )
> > and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by
> analogue, I
> > mean each cell looks like the respective puzzle and moves in a similar
> > manner]. In each of these puzzles, the new cut clips off the corners.
> > Remembering that to truly express the 4D nature of these puzzles, each
> cell
> > must be "exploded", so what used to be he vertex pieces for each of the=
se
> > puzzles have now been cut in half resulting in an internal piece that
> > behaves as one might have expected the single original piece to act and
> an
> > external piece that in addition to moving every time the internal piece
> > moves, can also be affected by a non-adjacent face.
> >
> >
> > As to a nice interface for non-face rotating 4D puzzles, my suggestion =
is
> to
> > display the wireframe of a 3D solid that displays all the symmetries
> implied
> > by the rotation between the faces and perform clicks not on the puzzle
> > itself, but only on this wireframe. For example, on a 4D Skewb, rotatio=
ns
> > are made around the "corners" of each cell. These rotations are all
> > equivalent to some rotation on a face turning 16Cell. So, in the
> Hypercube
> > shape, we could display wireframes of tetrahedrons that "float" between
> the
> > appropriate corners of 4 hypercube cells. When the user clicks on a fac=
e
> of
> > this floating wirefram tetrahedron, both the tetrahedron and the pieces
> > affected by the corresponding "vertex twist" all rotate. Clicking on th=
e
> > actual stickers of the puzzle does nothing; all rotations are executed =
by
> > clicking on these "rotation polyhedra". In the case of the 4D Helicopte=
r
> > Cube, the appropriate wireframe shape would be a triangular prism -
> > rotations around both the triangle faces and the rectangular faces are
> > possible moves on the 4D Helicopter Cube, and each of these rotations c=
an
> be
> > executed unambiguously by clicking on the appropriate face of the
> triangular
> > prism wireframe floating between the cells of the puzzle.
> >
> >
> > As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, that
> is -
> > a hypercube consisting of exploded skewbs (with additional cuts clippin=
g
> off
> > the corners). Now identify all the pieces affected by one particular
> > rotation and try to identify the move that is on the opposite side of t=
he
> > puzzle. Identified correctly, this opposite move does not affect any of
> the
> > same pieces. However, not every piece is affected by these two moves!
> There
> > is a band of pieces remaining untouched, much like the slice of a 3x3x3
> left
> > untouched by UD'. This means the puzzle is not deepcut! If we push the =
3D
> > hyper cutting planes deeper into the 4D puzzle, we get cells that look
> like
> > Master Skewbs. Continuing to push, certain pieces of these Master Skewb=
s
> get
> > thinner and thinner until they vanish at the point when opposing
> hyperplanes
> > meet. This is the deepcut vertex turning 8Cell puzzle. Each cell looks
> like
> > an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells
> that
> > look like dino cubes that is shallower cut. Although these puzzles are
> > visually identical, a single move on the shallower cut puzzle affects
> pieces
> > on only 4 cells while a single move on the deepcut puzzle affects piece=
s
> on
> > all 8 cells. Also of interest is the series of complicated looking
> puzzles
> > that appear at cut depths between the 4D Skewb and each of these dino
> cell
> > puzzles, although there are only 3 slices per axis in these puzzles (sa=
me
> > order as 3x3x3), each cell is an exploded Master Skewb!
> >
> > Although I have explored several other ideas, the three puzzles (4D FTO=
,
> 4D
> > Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
> > candidates for the next run of 4D puzzles, they implement some complex
> piece
> > interactions without becoming too large or too visually crowded.
> >
> > These puzzles are of an incredible interest to me, because the
> interactions
> > of the pieces are so much more intricate than the 120Cell or any of the
> > simplex vertex puzzles possible in the current MC4D program! As I
> mentioned
> > in my message to Roice, I have a good idea of how each of these puzzles
> look
> > and function and would gladly assist anyone (Roice? haha) who wants to
> > attempt to program it. In the meantime, I will take a look at the code
> Roice
> > has provided me and try to do some work myself, but I highly doubt I wi=
ll
> > have success without an experienced programmer's help ;)
> >
> > I would love to hear others' thoughts on these!
> > -Matt Galla
> >
>
>=20=20
>
--0015174c3fc2a7f24a049a89af54
Content-Type: text/html; charset=windows-1252
Content-Transfer-Encoding: quoted-printable
en the most "natural") extension of the=A0FTO to 4D is a cell tur=
ning 16Cell. However, I was looking for a puzzle where each cell looks like=
an FTO, and this obviously cannot be the case for a 16Cell, which has tetr=
ahedral faces.
some puzzles.
s like the 3D counterpart, in the case of all puzzles except tetrahedral an=
d icosahedral, this unambiguously assigns the 4D shape. In the case of a te=
trahedral puzzle, you can choose between the 5Cell, the 16Cell, and the 600=
Cell. In the case of icosahedral, this interpretation fails to produce an e=
quivalent puzzle.
d construct the equivalent 4D shape. In the case of the octahedron, 4 trian=
gles meet at a point (triangle being the 2-simplex). Thus the 4D equivalent=
should have 4 tetrahedra (tetrahedron being the 3-simplex)=A0meeting at an=
edge. This can unambiguously find analogues for all regular polyhedra (in =
the case of the FTO, this interpretation gives a 16Cell with pyraminx-like =
cells, the one schuma is referring to), and possibly more; however no puzzl=
e will ever get mapped to the 24Cell (because the 24Cell has no 3D equivale=
nt).
sense. You do not design a 3D puzzle by first deciding what each face shoul=
d look like and then repeating it over the rest of the puzzle. BUT YOU COUL=
D! ;) As long as you pick a face that is cut in such a way that all cuts ar=
e parallel to the sides of the face-shape and at equal depths, the resultin=
g puzzle should be "playable". (the 4D analogue for this is choos=
ing a cell layout such that all cuts are parallel to the faces of the cell =
and at equal depths - but this is PRECISELY what allows the cell to alone b=
e a 3D puzzle)
le some 4D puzzles can be obtained through either interpretation, there are=
some (like the 24Cell 4D FTO I described earlier) that can only be produce=
d through one interpretation. I therefore think it is important that we con=
sider both interpretations (plus I think a 24Cell would be more exciting, b=
ut maybe that's just me ;) ); PADDING-LEFT: 1ex" class=3D"gmail_quote">
Thank you for starting the discussions about other 4D pu=
zzles.
Can you explain more about why the 4D analogue of the FTO is=
a 24-cell instead of a 16-cell? Although the faces of the 24-cell are octa=
hedra, 24-cell is a self-dual polytope that is not a simplex. From this poi=
nt of view, it has no 3D analog. In fact it has no analog in any dimension =
other than 4D. However, the 16-cell belongs to the family of cross-polytope=
s, which are the duals of hypercubes, and exist in any number of dimensions=
. (">http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polyto=
pe is 16-cell. Therefore I think a natural extension of FTO is a cell-turni=
ng 16-cell, because they share more similarities.
For example, you may know that in 3D, the FTO can be regarded as a shap=
e-mod of Rex Cube, a vertex turning cube (es.com/forum/viewtopic.php?f=3D15&t=3D12659" target=3D"_blank">http://w=
ww.twistypuzzles.com/forum/viewtopic.php?f=3D15&t=3D12659). If the =
4D FTO is a shape-mod of the vertex turning hypercube, it should be a cell-=
turning 16-cell instead of a cell-turning 24-cell.
No matter calling it 4D FTO or else, I believe what you have described =
in the third paragraph is a cell-turning 24-cell. It should be an amazing p=
uzzle to solve. I have special feeling about it because of its uniqueness i=
n all the dimensions.
Nan=20
--- In com" target=3D"_blank">4D_Cubing@yahoogroups.com, "Galla, Matthew&=
quot; <mgalla@...> wrote:
>
> Hey everyone,
>
&=
gt; As I mentioned in my response about my solve of the 120Cell, I have bee=
n
> looking into some other 4D puzzles and have worked out how several of =
these
> puzzles should work and even discovered some interesting prop=
erties. Here is
> a snipet from my 120Cell solve message I sent Roice=
discussing this subject:
>
> "I am still hoping for more complicated 4D puzzles and a=
m willing to do
> whatever I can to help make them a reality. Coding =
a 4d space like you have
> is quite intimidating, but perhaps I can t=
ry to build off a pre-existing one
> with some guidance. I have already determined what the 4D analogue of =
the
> FTO (face turning octahedron, invented some time last year if y=
ou have not
> already seen it) would look like and how it would funct=
ion as well as the 4D
> analogue of the Skewb and Helicopter Cube (on that note I also have a<=
br>> suggestion as to how to make the interface for 4D puzzles that are =
non-face
> rotating, like the Skewb and Helicopter Cube). I have also=
made some
> interesting discoveries like for example making a 4D puzzle out of a 3=
D
> puzzle can make some additional internal cuts without altering th=
e exterior
> of a 3D face (true for all three puzzle I mentioned so f=
ar) and how a 4D
> Skewb is not deepcut! (that is every cell looks like a Skewb and seems=
to
> behave as such) The vertex turning deepcut hypercube has faces =
that
> externally each look like a dino cube. Is there anything I can=
do to make
> help make these a reality? After spending 150 hours on the 120Cell, I =
can
> honestly say that about 146 of the hours all feel exactly the s=
ame and I am
> dying to find a more interesting 4D puzzle to explore =
:)"
>
> To expand a little on some of the things I mentioned above, t=
he 4D FTO would
> be a 24Cell with faces that look like an exploded v=
ersion of this puzzle:
> taface.htm" target=3D"_blank">http://www.jaapsch.net/puzzles/octaface.htma>
> with one big difference, in addition to every cut on the 3D analogue o=
f the
> puzzle, the 4D version has and additional cut perpendicular t=
o the vertices
> of each face that line up with first cut down. :/ So=
rry, I know that wasn't
> very well worded and I'm not sure how well sending a picture would=
work
> through a yahoo group. Let me try again: these extra cuts wou=
ld essential
> cut off the vertex pieces of each cell. Removing the p=
ieces that are
> affected by this new unexpected cut will result in cells that have an<=
br>> exterior that matches this puzzle:
> puzzles.com/cgi-bin/puzzle.cgi?pkey=3D451" target=3D"_blank">http://twistyp=
uzzles.com/cgi-bin/puzzle.cgi?pkey=3D451
> (If you can follow my inadequate descriptions above, the 4D FTO would =
have 6
> distinct visible pieces, not just the 5 present on an explod=
ed 3D FTO - the
> extra comes from splitting each of the vertex piece=
s of the 3D Fto in half)
>
> A similar phenomenon occurs on both the 4D helicopter cube (3=
D:
> m_content&view=3Darticle&id=3D49:hcannounce&catid=3D1:latest-ne=
ws&Itemid=3D50" target=3D"_blank">http://www.puzzleforge.com/main/index=
.php?option=3Dcom_content&view=3Darticle&id=3D49:hcannounce&cat=
id=3D1:latest-news&Itemid=3D50)
> and 4D Skewb (3D: target=3D"_blank">http://www.jaapsch.net/puzzles/skewb.htm) [by analog=
ue, I
> mean each cell looks like the respective puzzle and moves in =
a similar
> manner]. In each of these puzzles, the new cut clips off the corners.<=
br>> Remembering that to truly express the 4D nature of these puzzles, e=
ach cell
> must be "exploded", so what used to be he vertex=
pieces for each of these
> puzzles have now been cut in half resulting in an internal piece that<=
br>> behaves as one might have expected the single original piece to act=
and an
> external piece that in addition to moving every time the in=
ternal piece
> moves, can also be affected by a non-adjacent face.
>
> <=
br>> As to a nice interface for non-face rotating 4D puzzles, my suggest=
ion is to
> display the wireframe of a 3D solid that displays all the=
symmetries implied
> by the rotation between the faces and perform clicks not on the puzzle=
> itself, but only on this wireframe. For example, on a 4D Skewb, ro=
tations
> are made around the "corners" of each cell. These=
rotations are all
> equivalent to some rotation on a face turning 16Cell. So, in the Hyper=
cube
> shape, we could display wireframes of tetrahedrons that "=
float" between the
> appropriate corners of 4 hypercube cells. W=
hen the user clicks on a face of
> this floating wirefram tetrahedron, both the tetrahedron and the piece=
s
> affected by the corresponding "vertex twist" all rotate=
. Clicking on the
> actual stickers of the puzzle does nothing; all r=
otations are executed by
> clicking on these "rotation polyhedra". In the case of the 4=
D Helicopter
> Cube, the appropriate wireframe shape would be a trian=
gular prism -
> rotations around both the triangle faces and the rect=
angular faces are
> possible moves on the 4D Helicopter Cube, and each of these rotations =
can be
> executed unambiguously by clicking on the appropriate face o=
f the triangular
> prism wireframe floating between the cells of the =
puzzle.
>
>
> As to the deepcut comment, attempt to visualize a 4D=
Skewb puzzle, that is -
> a hypercube consisting of exploded skewbs =
(with additional cuts clipping off
> the corners). Now identify all t=
he pieces affected by one particular
> rotation and try to identify the move that is on the opposite side of =
the
> puzzle. Identified correctly, this opposite move does not affec=
t any of the
> same pieces. However, not every piece is affected by t=
hese two moves! There
> is a band of pieces remaining untouched, much like the slice of a 3x3x=
3 left
> untouched by UD'. This means the puzzle is not deepcut! =
If we push the 3D
> hyper cutting planes deeper into the 4D puzzle, w=
e get cells that look like
> Master Skewbs. Continuing to push, certain pieces of these Master Skew=
bs get
> thinner and thinner until they vanish at the point when oppo=
sing hyperplanes
> meet. This is the deepcut vertex turning 8Cell puz=
zle. Each cell looks like
> an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells =
that
> look like dino cubes that is shallower cut. Although these puz=
zles are
> visually identical, a single move on the shallower cut puz=
zle affects pieces
> on only 4 cells while a single move on the deepcut puzzle affects piec=
es on
> all 8 cells. Also of interest is the series of complicated lo=
oking puzzles
> that appear at cut depths between the 4D Skewb and ea=
ch of these dino cell
> puzzles, although there are only 3 slices per axis in these puzzles (s=
ame
> order as 3x3x3), each cell is an exploded Master Skewb!
>=
> Although I have explored several other ideas, the three puzzles (=
4D FTO, 4D
> Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
=
> candidates for the next run of 4D puzzles, they implement some complex=
piece
> interactions without becoming too large or too visually crow=
ded.
>
> These puzzles are of an incredible interest to me, because th=
e interactions
> of the pieces are so much more intricate than the 12=
0Cell or any of the
> simplex vertex puzzles possible in the current =
MC4D program! As I mentioned
> in my message to Roice, I have a good idea of how each of these puzzle=
s look
> and function and would gladly assist anyone (Roice? haha) wh=
o wants to
> attempt to program it. In the meantime, I will take a lo=
ok at the code Roice
> has provided me and try to do some work myself, but I highly doubt I w=
ill
> have success without an experienced programmer's help ;)
> I would love to hear others' thoughts on these!
> =
-Matt Galla
>
--0015174c3fc2a7f24a049a89af54--
From: Roice Nelson <roice3@gmail.com>
Date: Tue, 25 Jan 2011 22:53:11 -0600
Subject: Re: [MC4D] Re: Other 4D puzzles
--001636c5a9ab27ae42049ab89c0d
Content-Type: text/plain; charset=ISO-8859-1
Great stuff guys!
Special thanks for helping me to picture the nature of a cell-turning
24-cell puzzle. In trying to understand the extra cuts you described, I see
now that they are somewhat related to the
incidence
of adjacent cells. In particular, the unusual cuts come from
the adjacent cells with vertex-only incidences. (btw, "parallel to vertex"
rather than "perpendicular to vertex" seems like decent language, though
this wording does refer to the adjacent cell the cut is based on.) At first
I thought the 24-cell puzzle would also need cuts parallel to edges, but
there are no adjacent cells having incident edges which do not also
have incident planes. On the 16-cell, there are adjacent cells with all
three possible incidence types, and it looks like there will be three styles
of cuts on its tetrahedral cells. Both puzzles sound difficult! We've
never run into these kinds of situations before because the adjacent cells
on puzzles with simplex vertex figures all have incident planes.
I also thought I'd mention that I never felt fully comfortable calling
Magic120Cell a "4D Megaminx", due to some of the analogy ambiguities you are
discussing. Similarly, my personal preference leans towards not using terms
like "4D Skewb", unless all could agree on the most defining Skewb-like
properties. Since a 3D Skewb is a vertex-turning puzzle with slices halfway
between diametrically opposed vertices, it could be argued that the 4D Skewb
must have all these properties, with the only change being that the
properties are now applied to a hypercube (in other words, that the 4D Skewb
is the puzzle you described that has faces that look like Dino cubes). I
guess my point is that I prefer language like Nan used, explicitly
describing the polytope and the nature of the twisting. But I also agree
the naming is not the most important aspect (and I've never been good at
creating interesting puzzle names), so that's all I will have to say about
that :)
Cheers,
Roice
P.S. Anyone know where you can buy the face-turning-octahedron puzzle? I'd
like to own one.
On Sun, Jan 23, 2011 at 2:54 PM, Galla, Matthew
>
>
> Ah,
>
> schuma (Nan?) is quite right. A very "natural" (perhaps even the most
> "natural") extension of the FTO to 4D is a cell turning 16Cell. However, I
> was looking for a puzzle where each cell looks like an FTO, and this
> obviously cannot be the case for a 16Cell, which has tetrahedral faces.
>
> It seems that in 4D there are two ways of interpreting the analogue of some
> puzzles.
> On the one hand, you could construct a 4D puzzle where every cell looks
> like the 3D counterpart, in the case of all puzzles except tetrahedral and
> icosahedral, this unambiguously assigns the 4D shape. In the case of a
> tetrahedral puzzle, you can choose between the 5Cell, the 16Cell, and the
> 600Cell. In the case of icosahedral, this interpretation fails to produce an
> equivalent puzzle.
> On the other hand, you can analyze the construction of the 3D shape and
> construct the equivalent 4D shape. In the case of the octahedron, 4
> triangles meet at a point (triangle being the 2-simplex). Thus the 4D
> equivalent should have 4 tetrahedra (tetrahedron being the
> 3-simplex) meeting at an edge. This can unambiguously find analogues for all
> regular polyhedra (in the case of the FTO, this interpretation gives a
> 16Cell with pyraminx-like cells, the one schuma is referring to), and
> possibly more; however no puzzle will ever get mapped to the 24Cell (because
> the 24Cell has no 3D equivalent).
>
> I realize the first method given above is "artificial" in a sense. You do
> not design a 3D puzzle by first deciding what each face should look like and
> then repeating it over the rest of the puzzle. BUT YOU COULD! ;) As long as
> you pick a face that is cut in such a way that all cuts are parallel to the
> sides of the face-shape and at equal depths, the resulting puzzle should be
> "playable". (the 4D analogue for this is choosing a cell layout such that
> all cuts are parallel to the faces of the cell and at equal depths - but
> this is PRECISELY what allows the cell to alone be a 3D puzzle)
>
> In any case, it seems that both methods produce valid puzzles, and while
> some 4D puzzles can be obtained through either interpretation, there are
> some (like the 24Cell 4D FTO I described earlier) that can only be produced
> through one interpretation. I therefore think it is important that we
> consider both interpretations (plus I think a 24Cell would be more exciting,
> but maybe that's just me ;) )
>
>
> Thanks for bringing that up schuma!
>
> -Matt Galla
>
> PS On TP my username is Allagem ;)
>
> On Sun, Jan 23, 2011 at 12:44 PM, schuma
>
>>
>>
>> Hi Matt,
>>
>> Thank you for starting the discussions about other 4D puzzles.
>>
>> Can you explain more about why the 4D analogue of the FTO is a 24-cell
>> instead of a 16-cell? Although the faces of the 24-cell are octahedra,
>> 24-cell is a self-dual polytope that is not a simplex. From this point of
>> view, it has no 3D analog. In fact it has no analog in any dimension other
>> than 4D. However, the 16-cell belongs to the family of cross-polytopes,
>> which are the duals of hypercubes, and exist in any number of dimensions. (
>> http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polytope
>> is 16-cell. Therefore I think a natural extension of FTO is a cell-turning
>> 16-cell, because they share more similarities.
>>
>> For example, you may know that in 3D, the FTO can be regarded as a
>> shape-mod of Rex Cube, a vertex turning cube (
>> http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=12659). If the 4D
>> FTO is a shape-mod of the vertex turning hypercube, it should be a
>> cell-turning 16-cell instead of a cell-turning 24-cell.
>>
>> No matter calling it 4D FTO or else, I believe what you have described in
>> the third paragraph is a cell-turning 24-cell. It should be an amazing
>> puzzle to solve. I have special feeling about it because of its uniqueness
>> in all the dimensions.
>>
>> Nan
>>
>>
>> --- In 4D_Cubing@yahoogroups.com <4D_Cubing%40yahoogroups.com>, "Galla,
>> Matthew"
>> >
>> > Hey everyone,
>> >
>> > As I mentioned in my response about my solve of the 120Cell, I have been
>> > looking into some other 4D puzzles and have worked out how several of
>> these
>> > puzzles should work and even discovered some interesting properties.
>> Here is
>> > a snipet from my 120Cell solve message I sent Roice discussing this
>> subject:
>> >
>> > "I am still hoping for more complicated 4D puzzles and am willing to do
>> > whatever I can to help make them a reality. Coding a 4d space like you
>> have
>> > is quite intimidating, but perhaps I can try to build off a pre-existing
>> one
>> > with some guidance. I have already determined what the 4D analogue of
>> the
>> > FTO (face turning octahedron, invented some time last year if you have
>> not
>> > already seen it) would look like and how it would function as well as
>> the 4D
>> > analogue of the Skewb and Helicopter Cube (on that note I also have a
>> > suggestion as to how to make the interface for 4D puzzles that are
>> non-face
>> > rotating, like the Skewb and Helicopter Cube). I have also made some
>> > interesting discoveries like for example making a 4D puzzle out of a 3D
>> > puzzle can make some additional internal cuts without altering the
>> exterior
>> > of a 3D face (true for all three puzzle I mentioned so far) and how a 4D
>> > Skewb is not deepcut! (that is every cell looks like a Skewb and seems
>> to
>> > behave as such) The vertex turning deepcut hypercube has faces that
>> > externally each look like a dino cube. Is there anything I can do to
>> make
>> > help make these a reality? After spending 150 hours on the 120Cell, I
>> can
>> > honestly say that about 146 of the hours all feel exactly the same and I
>> am
>> > dying to find a more interesting 4D puzzle to explore :)"
>> >
>> > To expand a little on some of the things I mentioned above, the 4D FTO
>> would
>> > be a 24Cell with faces that look like an exploded version of this
>> puzzle:
>> > http://www.jaapsch.net/puzzles/octaface.htm
>> > with one big difference, in addition to every cut on the 3D analogue of
>> the
>> > puzzle, the 4D version has and additional cut perpendicular to the
>> vertices
>> > of each face that line up with first cut down. :/ Sorry, I know that
>> wasn't
>> > very well worded and I'm not sure how well sending a picture would work
>> > through a yahoo group. Let me try again: these extra cuts would
>> essential
>> > cut off the vertex pieces of each cell. Removing the pieces that are
>> > affected by this new unexpected cut will result in cells that have an
>> > exterior that matches this puzzle:
>> > http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=451
>> > (If you can follow my inadequate descriptions above, the 4D FTO would
>> have 6
>> > distinct visible pieces, not just the 5 present on an exploded 3D FTO -
>> the
>> > extra comes from splitting each of the vertex pieces of the 3D Fto in
>> half)
>> >
>> > A similar phenomenon occurs on both the 4D helicopter cube (3D:
>> >
>> http://www.puzzleforge.com/main/index.php?option=com_content&view=article&id=49:hcannounce&catid=1:latest-news&Itemid=50
>> )
>> > and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by
>> analogue, I
>> > mean each cell looks like the respective puzzle and moves in a similar
>> > manner]. In each of these puzzles, the new cut clips off the corners.
>> > Remembering that to truly express the 4D nature of these puzzles, each
>> cell
>> > must be "exploded", so what used to be he vertex pieces for each of
>> these
>> > puzzles have now been cut in half resulting in an internal piece that
>> > behaves as one might have expected the single original piece to act and
>> an
>> > external piece that in addition to moving every time the internal piece
>> > moves, can also be affected by a non-adjacent face.
>> >
>> >
>> > As to a nice interface for non-face rotating 4D puzzles, my suggestion
>> is to
>> > display the wireframe of a 3D solid that displays all the symmetries
>> implied
>> > by the rotation between the faces and perform clicks not on the puzzle
>> > itself, but only on this wireframe. For example, on a 4D Skewb,
>> rotations
>> > are made around the "corners" of each cell. These rotations are all
>> > equivalent to some rotation on a face turning 16Cell. So, in the
>> Hypercube
>> > shape, we could display wireframes of tetrahedrons that "float" between
>> the
>> > appropriate corners of 4 hypercube cells. When the user clicks on a face
>> of
>> > this floating wirefram tetrahedron, both the tetrahedron and the pieces
>> > affected by the corresponding "vertex twist" all rotate. Clicking on the
>> > actual stickers of the puzzle does nothing; all rotations are executed
>> by
>> > clicking on these "rotation polyhedra". In the case of the 4D Helicopter
>> > Cube, the appropriate wireframe shape would be a triangular prism -
>> > rotations around both the triangle faces and the rectangular faces are
>> > possible moves on the 4D Helicopter Cube, and each of these rotations
>> can be
>> > executed unambiguously by clicking on the appropriate face of the
>> triangular
>> > prism wireframe floating between the cells of the puzzle.
>> >
>> >
>> > As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, that
>> is -
>> > a hypercube consisting of exploded skewbs (with additional cuts clipping
>> off
>> > the corners). Now identify all the pieces affected by one particular
>> > rotation and try to identify the move that is on the opposite side of
>> the
>> > puzzle. Identified correctly, this opposite move does not affect any of
>> the
>> > same pieces. However, not every piece is affected by these two moves!
>> There
>> > is a band of pieces remaining untouched, much like the slice of a 3x3x3
>> left
>> > untouched by UD'. This means the puzzle is not deepcut! If we push the
>> 3D
>> > hyper cutting planes deeper into the 4D puzzle, we get cells that look
>> like
>> > Master Skewbs. Continuing to push, certain pieces of these Master Skewbs
>> get
>> > thinner and thinner until they vanish at the point when opposing
>> hyperplanes
>> > meet. This is the deepcut vertex turning 8Cell puzzle. Each cell looks
>> like
>> > an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells
>> that
>> > look like dino cubes that is shallower cut. Although these puzzles are
>> > visually identical, a single move on the shallower cut puzzle affects
>> pieces
>> > on only 4 cells while a single move on the deepcut puzzle affects pieces
>> on
>> > all 8 cells. Also of interest is the series of complicated looking
>> puzzles
>> > that appear at cut depths between the 4D Skewb and each of these dino
>> cell
>> > puzzles, although there are only 3 slices per axis in these puzzles
>> (same
>> > order as 3x3x3), each cell is an exploded Master Skewb!
>> >
>> > Although I have explored several other ideas, the three puzzles (4D FTO,
>> 4D
>> > Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
>> > candidates for the next run of 4D puzzles, they implement some complex
>> piece
>> > interactions without becoming too large or too visually crowded.
>> >
>> > These puzzles are of an incredible interest to me, because the
>> interactions
>> > of the pieces are so much more intricate than the 120Cell or any of the
>> > simplex vertex puzzles possible in the current MC4D program! As I
>> mentioned
>> > in my message to Roice, I have a good idea of how each of these puzzles
>> look
>> > and function and would gladly assist anyone (Roice? haha) who wants to
>> > attempt to program it. In the meantime, I will take a look at the code
>> Roice
>> > has provided me and try to do some work myself, but I highly doubt I
>> will
>> > have success without an experienced programmer's help ;)
>> >
>> > I would love to hear others' thoughts on these!
>> > -Matt Galla
>> >
>>
>>
>
>
>
--001636c5a9ab27ae42049ab89c0d
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable
Great stuff guys!=20
24-cell puzzle.=A0 In trying to understand=A0the extra cuts you described, =
I see now that they are somewhat related to the dia.org/wiki/Incidence_(geometry)" target=3D"_blank">incidence properti=
es of adjacent cells.=A0 In particular, the unusual=A0cuts come from the=A0=
adjacent cells=A0with vertex-only incidences.=A0(btw, "parallel to ver=
tex" rather than "perpendicular to vertex" seems like=A0dece=
nt language, though this wording does refer to the adjacent cell the cut is=
based on.) =A0At first I thought the 24-cell puzzle would also need cuts p=
arallel to edges, but there are no adjacent cells having incident edges whi=
ch do not also have=A0incident planes. =A0On the 16-cell, there are adjacen=
t cells with all three possible incidence types, and it looks like there wi=
ll be=A0three styles of cuts on its tetrahedral cells. =A0Both puzzles soun=
d difficult!=A0 We've never run into these kinds of situations before b=
ecause the adjacent cells on puzzles with simplex vertex figures all have i=
ncident planes.
ling Magic120Cell a "4D Megaminx", due to some of the analogy amb=
iguities you are discussing. =A0Similarly, my personal preference leans tow=
ards not using terms like "4D Skewb", unless all could agree on t=
he most defining Skewb-like properties. =A0Since a 3D Skewb is a vertex-tur=
ning puzzle with slices halfway between diametrically opposed vertices, it =
could be argued that the 4D Skewb must have all these properties, with the =
only change being that the properties are now applied to a hypercube (in ot=
her words, that the 4D Skewb is the puzzle you described that has faces tha=
t look like Dino cubes). =A0I guess my point is that I prefer language like=
Nan used, explicitly describing the polytope and the nature of the twistin=
g. =A0But I also agree the naming is not the most important aspect (and I&#=
39;ve never been good at creating interesting puzzle names), so that's =
all I will have to say about that :)
face-turning-octahedron puzzle? =A0I'd like to own one.
<k">mgalla@trinity.edu> wrote:dding-left:1ex" class=3D"gmail_quote">
en the most "natural") extension of the=A0FTO to 4D is a cell tur=
ning 16Cell. However, I was looking for a puzzle where each cell looks like=
an FTO, and this obviously cannot be the case for a 16Cell, which has tetr=
ahedral faces.
some puzzles.
s like the 3D counterpart, in the case of all puzzles except tetrahedral an=
d icosahedral, this unambiguously assigns the 4D shape. In the case of a te=
trahedral puzzle, you can choose between the 5Cell, the 16Cell, and the 600=
Cell. In the case of icosahedral, this interpretation fails to produce an e=
quivalent puzzle.
d construct the equivalent 4D shape. In the case of the octahedron, 4 trian=
gles meet at a point (triangle being the 2-simplex). Thus the 4D equivalent=
should have 4 tetrahedra (tetrahedron being the 3-simplex)=A0meeting at an=
edge. This can unambiguously find analogues for all regular polyhedra (in =
the case of the FTO, this interpretation gives a 16Cell with pyraminx-like =
cells, the one schuma is referring to), and possibly more; however no puzzl=
e will ever get mapped to the 24Cell (because the 24Cell has no 3D equivale=
nt).
sense. You do not design a 3D puzzle by first deciding what each face shoul=
d look like and then repeating it over the rest of the puzzle. BUT YOU COUL=
D! ;) As long as you pick a face that is cut in such a way that all cuts ar=
e parallel to the sides of the face-shape and at equal depths, the resultin=
g puzzle should be "playable". (the 4D analogue for this is choos=
ing a cell layout such that all cuts are parallel to the faces of the cell =
and at equal depths - but this is PRECISELY what allows the cell to alone b=
e a 3D puzzle)
le some 4D puzzles can be obtained through either interpretation, there are=
some (like the 24Cell 4D FTO I described earlier) that can only be produce=
d through one interpretation. I therefore think it is important that we con=
sider both interpretations (plus I think a 24Cell would be more exciting, b=
ut maybe that's just me ;) )
anself@gmail.com> wrote:dding-left:1ex" class=3D"gmail_quote">
Thank you for starting the discussions about other 4D pu=
zzles.
Can you explain more about why the 4D analogue of the FTO is=
a 24-cell instead of a 16-cell? Although the faces of the 24-cell are octa=
hedra, 24-cell is a self-dual polytope that is not a simplex. From this poi=
nt of view, it has no 3D analog. In fact it has no analog in any dimension =
other than 4D. However, the 16-cell belongs to the family of cross-polytope=
s, which are the duals of hypercubes, and exist in any number of dimensions=
. (">http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polyto=
pe is 16-cell. Therefore I think a natural extension of FTO is a cell-turni=
ng 16-cell, because they share more similarities.
For example, you may know that in 3D, the FTO can be regarded as a shap=
e-mod of Rex Cube, a vertex turning cube (es.com/forum/viewtopic.php?f=3D15&t=3D12659" target=3D"_blank">http://w=
ww.twistypuzzles.com/forum/viewtopic.php?f=3D15&t=3D12659). If the =
4D FTO is a shape-mod of the vertex turning hypercube, it should be a cell-=
turning 16-cell instead of a cell-turning 24-cell.
No matter calling it 4D FTO or else, I believe what you have described =
in the third paragraph is a cell-turning 24-cell. It should be an amazing p=
uzzle to solve. I have special feeling about it because of its uniqueness i=
n all the dimensions.
Nan
--- In =3D"_blank">4D_Cubing@yahoogroups.com, "Galla, Matthew" <m=
galla@...> wrote:
>
> Hey everyone,
>
> As I me=
ntioned in my response about my solve of the 120Cell, I have been
> looking into some other 4D puzzles and have worked out how several of =
these
> puzzles should work and even discovered some interesting prop=
erties. Here is
> a snipet from my 120Cell solve message I sent Roice=
discussing this subject:
>
> "I am still hoping for more complicated 4D puzzles and a=
m willing to do
> whatever I can to help make them a reality. Coding =
a 4d space like you have
> is quite intimidating, but perhaps I can t=
ry to build off a pre-existing one
> with some guidance. I have already determined what the 4D analogue of =
the
> FTO (face turning octahedron, invented some time last year if y=
ou have not
> already seen it) would look like and how it would funct=
ion as well as the 4D
> analogue of the Skewb and Helicopter Cube (on that note I also have a<=
br>> suggestion as to how to make the interface for 4D puzzles that are =
non-face
> rotating, like the Skewb and Helicopter Cube). I have also=
made some
> interesting discoveries like for example making a 4D puzzle out of a 3=
D
> puzzle can make some additional internal cuts without altering th=
e exterior
> of a 3D face (true for all three puzzle I mentioned so f=
ar) and how a 4D
> Skewb is not deepcut! (that is every cell looks like a Skewb and seems=
to
> behave as such) The vertex turning deepcut hypercube has faces =
that
> externally each look like a dino cube. Is there anything I can=
do to make
> help make these a reality? After spending 150 hours on the 120Cell, I =
can
> honestly say that about 146 of the hours all feel exactly the s=
ame and I am
> dying to find a more interesting 4D puzzle to explore =
:)"
>
> To expand a little on some of the things I mentioned above, t=
he 4D FTO would
> be a 24Cell with faces that look like an exploded v=
ersion of this puzzle:
> taface.htm" target=3D"_blank">http://www.jaapsch.net/puzzles/octaface.htma>
> with one big difference, in addition to every cut on the 3D analogue o=
f the
> puzzle, the 4D version has and additional cut perpendicular t=
o the vertices
> of each face that line up with first cut down. :/ So=
rry, I know that wasn't
> very well worded and I'm not sure how well sending a picture would=
work
> through a yahoo group. Let me try again: these extra cuts wou=
ld essential
> cut off the vertex pieces of each cell. Removing the p=
ieces that are
> affected by this new unexpected cut will result in cells that have an<=
br>> exterior that matches this puzzle:
> puzzles.com/cgi-bin/puzzle.cgi?pkey=3D451" target=3D"_blank">http://twistyp=
uzzles.com/cgi-bin/puzzle.cgi?pkey=3D451
> (If you can follow my inadequate descriptions above, the 4D FTO would =
have 6
> distinct visible pieces, not just the 5 present on an explod=
ed 3D FTO - the
> extra comes from splitting each of the vertex piece=
s of the 3D Fto in half)
>
> A similar phenomenon occurs on both the 4D helicopter cube (3=
D:
> m_content&view=3Darticle&id=3D49:hcannounce&catid=3D1:latest-ne=
ws&Itemid=3D50" target=3D"_blank">http://www.puzzleforge.com/main/index=
.php?option=3Dcom_content&view=3Darticle&id=3D49:hcannounce&cat=
id=3D1:latest-news&Itemid=3D50)
> and 4D Skewb (3D: target=3D"_blank">http://www.jaapsch.net/puzzles/skewb.htm) [by analog=
ue, I
> mean each cell looks like the respective puzzle and moves in =
a similar
> manner]. In each of these puzzles, the new cut clips off the corners.<=
br>> Remembering that to truly express the 4D nature of these puzzles, e=
ach cell
> must be "exploded", so what used to be he vertex=
pieces for each of these
> puzzles have now been cut in half resulting in an internal piece that<=
br>> behaves as one might have expected the single original piece to act=
and an
> external piece that in addition to moving every time the in=
ternal piece
> moves, can also be affected by a non-adjacent face.
>
> <=
br>> As to a nice interface for non-face rotating 4D puzzles, my suggest=
ion is to
> display the wireframe of a 3D solid that displays all the=
symmetries implied
> by the rotation between the faces and perform clicks not on the puzzle=
> itself, but only on this wireframe. For example, on a 4D Skewb, ro=
tations
> are made around the "corners" of each cell. These=
rotations are all
> equivalent to some rotation on a face turning 16Cell. So, in the Hyper=
cube
> shape, we could display wireframes of tetrahedrons that "=
float" between the
> appropriate corners of 4 hypercube cells. W=
hen the user clicks on a face of
> this floating wirefram tetrahedron, both the tetrahedron and the piece=
s
> affected by the corresponding "vertex twist" all rotate=
. Clicking on the
> actual stickers of the puzzle does nothing; all r=
otations are executed by
> clicking on these "rotation polyhedra". In the case of the 4=
D Helicopter
> Cube, the appropriate wireframe shape would be a trian=
gular prism -
> rotations around both the triangle faces and the rect=
angular faces are
> possible moves on the 4D Helicopter Cube, and each of these rotations =
can be
> executed unambiguously by clicking on the appropriate face o=
f the triangular
> prism wireframe floating between the cells of the =
puzzle.
>
>
> As to the deepcut comment, attempt to visualize a 4D=
Skewb puzzle, that is -
> a hypercube consisting of exploded skewbs =
(with additional cuts clipping off
> the corners). Now identify all t=
he pieces affected by one particular
> rotation and try to identify the move that is on the opposite side of =
the
> puzzle. Identified correctly, this opposite move does not affec=
t any of the
> same pieces. However, not every piece is affected by t=
hese two moves! There
> is a band of pieces remaining untouched, much like the slice of a 3x3x=
3 left
> untouched by UD'. This means the puzzle is not deepcut! =
If we push the 3D
> hyper cutting planes deeper into the 4D puzzle, w=
e get cells that look like
> Master Skewbs. Continuing to push, certain pieces of these Master Skew=
bs get
> thinner and thinner until they vanish at the point when oppo=
sing hyperplanes
> meet. This is the deepcut vertex turning 8Cell puz=
zle. Each cell looks like
> an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells =
that
> look like dino cubes that is shallower cut. Although these puz=
zles are
> visually identical, a single move on the shallower cut puz=
zle affects pieces
> on only 4 cells while a single move on the deepcut puzzle affects piec=
es on
> all 8 cells. Also of interest is the series of complicated lo=
oking puzzles
> that appear at cut depths between the 4D Skewb and ea=
ch of these dino cell
> puzzles, although there are only 3 slices per axis in these puzzles (s=
ame
> order as 3x3x3), each cell is an exploded Master Skewb!
>=
> Although I have explored several other ideas, the three puzzles (=
4D FTO, 4D
> Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
=
> candidates for the next run of 4D puzzles, they implement some complex=
piece
> interactions without becoming too large or too visually crow=
ded.
>
> These puzzles are of an incredible interest to me, because th=
e interactions
> of the pieces are so much more intricate than the 12=
0Cell or any of the
> simplex vertex puzzles possible in the current =
MC4D program! As I mentioned
> in my message to Roice, I have a good idea of how each of these puzzle=
s look
> and function and would gladly assist anyone (Roice? haha) wh=
o wants to
> attempt to program it. In the meantime, I will take a lo=
ok at the code Roice
> has provided me and try to do some work myself, but I highly doubt I w=
ill
> have success without an experienced programmer's help ;)
> I would love to hear others' thoughts on these!
> -Matt Galla
>
--001636c5a9ab27ae42049ab89c0d--
From: Melinda Green <melinda@superliminal.com>
Date: Tue, 25 Jan 2011 21:47:36 -0800
Subject: Re: [MC4D] Re: Other 4D puzzles
--------------090305000108080706090004
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
On 1/25/2011 8:53 PM, Roice Nelson wrote:
>
>
> Great stuff guys!
>
> Special thanks for helping me to picture the nature of a cell-turning
> 24-cell puzzle. In trying to understand the extra cuts you described,
> I see now that they are somewhat related to the incidence
>
> adjacent cells. In particular, the unusual cuts come from
> the adjacent cells with vertex-only incidences. (btw, "parallel to
> vertex" rather than "perpendicular to vertex" seems like decent
> language, though this wording does refer to the adjacent cell the cut
> is based on.) At first I thought the 24-cell puzzle would also need
> cuts parallel to edges, but there are no adjacent cells having
> incident edges which do not also have incident planes. On the
> 16-cell, there are adjacent cells with all three possible incidence
> types, and it looks like there will be three styles of cuts on its
> tetrahedral cells. Both puzzles sound difficult! We've never run
> into these kinds of situations before because the adjacent cells on
> puzzles with simplex vertex figures all have incident planes.
Don's new engine may support the 24 cell but I'm not sure. It'd be good
to check before attempting an implementation.
>
> I also thought I'd mention that I never felt fully comfortable calling
> Magic120Cell a "4D Megaminx", due to some of the analogy ambiguities
> you are discussing.
Is there any other puzzle you can imagine that might deserve that
description? Either way we can certainly say that the Magic120Cell can
easily be considered to be the natural 4D analog of the 3D Megaminx.
That's my feeling anyway.
> Similarly, my personal preference leans towards not using terms like
> "4D Skewb", unless all could agree on the most defining Skewb-like
> properties. Since a 3D Skewb is a vertex-turning puzzle with slices
> halfway between diametrically opposed vertices, it could be argued
> that the 4D Skewb must have all these properties, with the only change
> being that the properties are now applied to a hypercube (in other
> words, that the 4D Skewb is the puzzle you described that has faces
> that look like Dino cubes). I guess my point is that I prefer
> language like Nan used, explicitly describing the polytope and the
> nature of the twisting. But I also agree the naming is not the most
> important aspect (and I've never been good at creating interesting
> puzzle names), so that's all I will have to say about that :)
Well I share your aesthetic of preferring to let form follow function
here rather than the other way around. I just hope that I'll be forgiven
if I'm contradicting what I just said about the Megaminx. :-)
>
> P.S. Anyone know where you can buy the face-turning-octahedron puzzle?
> I'd like to own one.
Me too! I don't know if I ever mentioned this but long, LONG ago within
a year or two of Erno Rubik's puzzle hit the planet like a virulent
meme, I decided to build a face-turning octahedra. I developed the
ability to combine plaster casts with silicone gel to handle some
concavities in order to cast fiberglass resin pieces. It was such slow
work that I never finished more than about a third of the pieces and a
central hub consisting of bolts with their heads welded to each other.
Not so long ago I decided I was never going to finish it so I threw most
of it away though I think that I kept a few of the cast pieces as they
were quite a lovely translucent amber. I've been waiting all these years
expecting it to pop up at some point but it never seemed like anyone was
going to make it. I was glad to see it on GelatinBrain however.
-Melinda
P.S. Please excuse that I truncated the thread history here as it was
getting very long and is archived in the Yahoo group. This is definitely
an interesting thread with lots more to say on the subject.
--------------090305000108080706090004
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit
http-equiv="Content-Type">
On 1/25/2011 8:53 PM, Roice Nelson wrote:
type="cite">
Great stuff guys!
cell-turning 24-cell puzzle. In trying to understand the extra
cuts you described, I see now that they are somewhat related to
the href="http://en.wikipedia.org/wiki/Incidence_%28geometry%29"
target="_blank">incidence properties of adjacent cells.
In particular, the unusual cuts come from the adjacent
cells with vertex-only incidences. (btw, "parallel to vertex"
rather than "perpendicular to vertex" seems like decent
language, though this wording does refer to the adjacent cell
the cut is based on.) At first I thought the 24-cell puzzle
would also need cuts parallel to edges, but there are no
adjacent cells having incident edges which do not also
have incident planes. On the 16-cell, there are adjacent cells
with all three possible incidence types, and it looks like there
will be three styles of cuts on its tetrahedral cells. Both
puzzles sound difficult! We've never run into these kinds of
situations before because the adjacent cells on puzzles with
simplex vertex figures all have incident planes.
Don's new engine may support the 24 cell but I'm not sure. It'd be
good to check before attempting an implementation.
type="cite">
comfortable calling Magic120Cell a "4D Megaminx", due to some of
the analogy ambiguities you are discussing.
Is there any other puzzle you can imagine that might deserve that
description? Either way we can certainly say that the Magic120Cell
can easily be considered to be the natural 4D analog of the 3D
Megaminx. That's my feeling anyway.
type="cite">
terms like "4D Skewb", unless all could agree on the most
defining Skewb-like properties. Since a 3D Skewb is a
vertex-turning puzzle with slices halfway between diametrically
opposed vertices, it could be argued that the 4D Skewb must have
all these properties, with the only change being that the
properties are now applied to a hypercube (in other words, that
the 4D Skewb is the puzzle you described that has faces that
look like Dino cubes). I guess my point is that I prefer
language like Nan used, explicitly describing the polytope and
the nature of the twisting. But I also agree the naming is not
the most important aspect (and I've never been good at creating
interesting puzzle names), so that's all I will have to say
about that :)
Well I share your aesthetic of preferring to let form follow
function here rather than the other way around. I just hope that
I'll be forgiven if I'm contradicting what I just said about the
Megaminx. :-)
type="cite">
face-turning-octahedron puzzle? I'd like to own one.
Me too! I don't know if I ever mentioned this but long, LONG ago
within a year or two of Erno Rubik's puzzle hit the planet like a
virulent meme, I decided to build a face-turning octahedra. I
developed the ability to combine plaster casts with silicone gel to
handle some concavities in order to cast fiberglass resin pieces. It
was such slow work that I never finished more than about a third of
the pieces and a central hub consisting of bolts with their heads
welded to each other. Not so long ago I decided I was never going to
finish it so I threw most of it away though I think that I kept a
few of the cast pieces as they were quite a lovely translucent
amber. I've been waiting all these years expecting it to pop up at
some point but it never seemed like anyone was going to make it. I
was glad to see it on GelatinBrain however.
-Melinda
P.S. Please excuse that I truncated the thread history here as it
was getting very long and is archived in the Yahoo group. This is
definitely an interesting thread with lots more to say on the
subject.
--------------090305000108080706090004--
From: "schuma" <mananself@gmail.com>
Date: Wed, 26 Jan 2011 05:55:13 -0000
Subject: [MC4D] Re: Other 4D puzzles
Hi,
This message is dedicated to answer Roice's question about where to buy a f=
ace turning octahedron. Here are some suggestions:
http://www.hknowstore.com/item.aspx?corpname=3Dnowstore&itemid=3D058fbd01-4=
a44-4e6f-99ec-71ae3bd9eb23
http://www.witeden.com/goods.php?id=3D174
or ebay sellers, for example
http://cgi.ebay.com/Magic-Octahedron-Rubiks-Cube-Star-Puzzler-Transparent-/=
160515420233?pt=3DLH_DefaultDomain_2&hash=3Ditem255f76f049
This puzzle has been mass-produced twice, so they are pretty cheap now, for=
around $10+shipping. Shipping might take two weeks because the sellers are=
usually in Hong Kong or China. Just make sure it's a face-turning one befo=
re you buy it, because the vertex turning octahedron is also mass produced,=
which looks almost the same.
Nan
--- In 4D_Cubing@yahoogroups.com, Roice Nelson
>
> Great stuff guys!
>=20
> Special thanks for helping me to picture the nature of a cell-turning
> 24-cell puzzle. In trying to understand the extra cuts you described, I =
see
> now that they are somewhat related to the
> incidence
> of adjacent cells. In particular, the unusual cuts come from
> the adjacent cells with vertex-only incidences. (btw, "parallel to vertex=
"
> rather than "perpendicular to vertex" seems like decent language, though
> this wording does refer to the adjacent cell the cut is based on.) At fi=
rst
> I thought the 24-cell puzzle would also need cuts parallel to edges, but
> there are no adjacent cells having incident edges which do not also
> have incident planes. On the 16-cell, there are adjacent cells with all
> three possible incidence types, and it looks like there will be three sty=
les
> of cuts on its tetrahedral cells. Both puzzles sound difficult! We've
> never run into these kinds of situations before because the adjacent cell=
s
> on puzzles with simplex vertex figures all have incident planes.
>=20
> I also thought I'd mention that I never felt fully comfortable calling
> Magic120Cell a "4D Megaminx", due to some of the analogy ambiguities you =
are
> discussing. Similarly, my personal preference leans towards not using te=
rms
> like "4D Skewb", unless all could agree on the most defining Skewb-like
> properties. Since a 3D Skewb is a vertex-turning puzzle with slices half=
way
> between diametrically opposed vertices, it could be argued that the 4D Sk=
ewb
> must have all these properties, with the only change being that the
> properties are now applied to a hypercube (in other words, that the 4D Sk=
ewb
> is the puzzle you described that has faces that look like Dino cubes). I
> guess my point is that I prefer language like Nan used, explicitly
> describing the polytope and the nature of the twisting. But I also agree
> the naming is not the most important aspect (and I've never been good at
> creating interesting puzzle names), so that's all I will have to say abou=
t
> that :)
>=20
> Cheers,
> Roice
>=20
> P.S. Anyone know where you can buy the face-turning-octahedron puzzle? I=
'd
> like to own one.
>=20
>=20
> On Sun, Jan 23, 2011 at 2:54 PM, Galla, Matthew
>=20
> >
> >
> > Ah,
> >
> > schuma (Nan?) is quite right. A very "natural" (perhaps even the most
> > "natural") extension of the FTO to 4D is a cell turning 16Cell. However=
, I
> > was looking for a puzzle where each cell looks like an FTO, and this
> > obviously cannot be the case for a 16Cell, which has tetrahedral faces.
> >
> > It seems that in 4D there are two ways of interpreting the analogue of =
some
> > puzzles.
> > On the one hand, you could construct a 4D puzzle where every cell looks
> > like the 3D counterpart, in the case of all puzzles except tetrahedral =
and
> > icosahedral, this unambiguously assigns the 4D shape. In the case of a
> > tetrahedral puzzle, you can choose between the 5Cell, the 16Cell, and t=
he
> > 600Cell. In the case of icosahedral, this interpretation fails to produ=
ce an
> > equivalent puzzle.
> > On the other hand, you can analyze the construction of the 3D shape and
> > construct the equivalent 4D shape. In the case of the octahedron, 4
> > triangles meet at a point (triangle being the 2-simplex). Thus the 4D
> > equivalent should have 4 tetrahedra (tetrahedron being the
> > 3-simplex) meeting at an edge. This can unambiguously find analogues fo=
r all
> > regular polyhedra (in the case of the FTO, this interpretation gives a
> > 16Cell with pyraminx-like cells, the one schuma is referring to), and
> > possibly more; however no puzzle will ever get mapped to the 24Cell (be=
cause
> > the 24Cell has no 3D equivalent).
> >
> > I realize the first method given above is "artificial" in a sense. You =
do
> > not design a 3D puzzle by first deciding what each face should look lik=
e and
> > then repeating it over the rest of the puzzle. BUT YOU COULD! ;) As lon=
g as
> > you pick a face that is cut in such a way that all cuts are parallel to=
the
> > sides of the face-shape and at equal depths, the resulting puzzle shoul=
d be
> > "playable". (the 4D analogue for this is choosing a cell layout such th=
at
> > all cuts are parallel to the faces of the cell and at equal depths - bu=
t
> > this is PRECISELY what allows the cell to alone be a 3D puzzle)
> >
> > In any case, it seems that both methods produce valid puzzles, and whil=
e
> > some 4D puzzles can be obtained through either interpretation, there ar=
e
> > some (like the 24Cell 4D FTO I described earlier) that can only be prod=
uced
> > through one interpretation. I therefore think it is important that we
> > consider both interpretations (plus I think a 24Cell would be more exci=
ting,
> > but maybe that's just me ;) )
> >
> >
> > Thanks for bringing that up schuma!
> >
> > -Matt Galla
> >
> > PS On TP my username is Allagem ;)
> >
> > On Sun, Jan 23, 2011 at 12:44 PM, schuma
> >
> >>
> >>
> >> Hi Matt,
> >>
> >> Thank you for starting the discussions about other 4D puzzles.
> >>
> >> Can you explain more about why the 4D analogue of the FTO is a 24-cell
> >> instead of a 16-cell? Although the faces of the 24-cell are octahedra,
> >> 24-cell is a self-dual polytope that is not a simplex. From this point=
of
> >> view, it has no 3D analog. In fact it has no analog in any dimension o=
ther
> >> than 4D. However, the 16-cell belongs to the family of cross-polytopes=
,
> >> which are the duals of hypercubes, and exist in any number of dimensio=
ns. (
> >> http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polytop=
e
> >> is 16-cell. Therefore I think a natural extension of FTO is a cell-tur=
ning
> >> 16-cell, because they share more similarities.
> >>
> >> For example, you may know that in 3D, the FTO can be regarded as a
> >> shape-mod of Rex Cube, a vertex turning cube (
> >> http://www.twistypuzzles.com/forum/viewtopic.php?f=3D15&t=3D12659). If=
the 4D
> >> FTO is a shape-mod of the vertex turning hypercube, it should be a
> >> cell-turning 16-cell instead of a cell-turning 24-cell.
> >>
> >> No matter calling it 4D FTO or else, I believe what you have described=
in
> >> the third paragraph is a cell-turning 24-cell. It should be an amazing
> >> puzzle to solve. I have special feeling about it because of its unique=
ness
> >> in all the dimensions.
> >>
> >> Nan
> >>
> >>
> >> --- In 4D_Cubing@yahoogroups.com <4D_Cubing%40yahoogroups.com>, "Galla=
,
> >> Matthew"
> >> >
> >> > Hey everyone,
> >> >
> >> > As I mentioned in my response about my solve of the 120Cell, I have =
been
> >> > looking into some other 4D puzzles and have worked out how several o=
f
> >> these
> >> > puzzles should work and even discovered some interesting properties.
> >> Here is
> >> > a snipet from my 120Cell solve message I sent Roice discussing this
> >> subject:
> >> >
> >> > "I am still hoping for more complicated 4D puzzles and am willing to=
do
> >> > whatever I can to help make them a reality. Coding a 4d space like y=
ou
> >> have
> >> > is quite intimidating, but perhaps I can try to build off a pre-exis=
ting
> >> one
> >> > with some guidance. I have already determined what the 4D analogue o=
f
> >> the
> >> > FTO (face turning octahedron, invented some time last year if you ha=
ve
> >> not
> >> > already seen it) would look like and how it would function as well a=
s
> >> the 4D
> >> > analogue of the Skewb and Helicopter Cube (on that note I also have =
a
> >> > suggestion as to how to make the interface for 4D puzzles that are
> >> non-face
> >> > rotating, like the Skewb and Helicopter Cube). I have also made some
> >> > interesting discoveries like for example making a 4D puzzle out of a=
3D
> >> > puzzle can make some additional internal cuts without altering the
> >> exterior
> >> > of a 3D face (true for all three puzzle I mentioned so far) and how =
a 4D
> >> > Skewb is not deepcut! (that is every cell looks like a Skewb and see=
ms
> >> to
> >> > behave as such) The vertex turning deepcut hypercube has faces that
> >> > externally each look like a dino cube. Is there anything I can do to
> >> make
> >> > help make these a reality? After spending 150 hours on the 120Cell, =
I
> >> can
> >> > honestly say that about 146 of the hours all feel exactly the same a=
nd I
> >> am
> >> > dying to find a more interesting 4D puzzle to explore :)"
> >> >
> >> > To expand a little on some of the things I mentioned above, the 4D F=
TO
> >> would
> >> > be a 24Cell with faces that look like an exploded version of this
> >> puzzle:
> >> > http://www.jaapsch.net/puzzles/octaface.htm
> >> > with one big difference, in addition to every cut on the 3D analogue=
of
> >> the
> >> > puzzle, the 4D version has and additional cut perpendicular to the
> >> vertices
> >> > of each face that line up with first cut down. :/ Sorry, I know that
> >> wasn't
> >> > very well worded and I'm not sure how well sending a picture would w=
ork
> >> > through a yahoo group. Let me try again: these extra cuts would
> >> essential
> >> > cut off the vertex pieces of each cell. Removing the pieces that are
> >> > affected by this new unexpected cut will result in cells that have a=
n
> >> > exterior that matches this puzzle:
> >> > http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=3D451
> >> > (If you can follow my inadequate descriptions above, the 4D FTO woul=
d
> >> have 6
> >> > distinct visible pieces, not just the 5 present on an exploded 3D FT=
O -
> >> the
> >> > extra comes from splitting each of the vertex pieces of the 3D Fto i=
n
> >> half)
> >> >
> >> > A similar phenomenon occurs on both the 4D helicopter cube (3D:
> >> >
> >> http://www.puzzleforge.com/main/index.php?option=3Dcom_content&view=3D=
article&id=3D49:hcannounce&catid=3D1:latest-news&Itemid=3D50
> >> )
> >> > and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by
> >> analogue, I
> >> > mean each cell looks like the respective puzzle and moves in a simil=
ar
> >> > manner]. In each of these puzzles, the new cut clips off the corners=
.
> >> > Remembering that to truly express the 4D nature of these puzzles, ea=
ch
> >> cell
> >> > must be "exploded", so what used to be he vertex pieces for each of
> >> these
> >> > puzzles have now been cut in half resulting in an internal piece tha=
t
> >> > behaves as one might have expected the single original piece to act =
and
> >> an
> >> > external piece that in addition to moving every time the internal pi=
ece
> >> > moves, can also be affected by a non-adjacent face.
> >> >
> >> >
> >> > As to a nice interface for non-face rotating 4D puzzles, my suggesti=
on
> >> is to
> >> > display the wireframe of a 3D solid that displays all the symmetries
> >> implied
> >> > by the rotation between the faces and perform clicks not on the puzz=
le
> >> > itself, but only on this wireframe. For example, on a 4D Skewb,
> >> rotations
> >> > are made around the "corners" of each cell. These rotations are all
> >> > equivalent to some rotation on a face turning 16Cell. So, in the
> >> Hypercube
> >> > shape, we could display wireframes of tetrahedrons that "float" betw=
een
> >> the
> >> > appropriate corners of 4 hypercube cells. When the user clicks on a =
face
> >> of
> >> > this floating wirefram tetrahedron, both the tetrahedron and the pie=
ces
> >> > affected by the corresponding "vertex twist" all rotate. Clicking on=
the
> >> > actual stickers of the puzzle does nothing; all rotations are execut=
ed
> >> by
> >> > clicking on these "rotation polyhedra". In the case of the 4D Helico=
pter
> >> > Cube, the appropriate wireframe shape would be a triangular prism -
> >> > rotations around both the triangle faces and the rectangular faces a=
re
> >> > possible moves on the 4D Helicopter Cube, and each of these rotation=
s
> >> can be
> >> > executed unambiguously by clicking on the appropriate face of the
> >> triangular
> >> > prism wireframe floating between the cells of the puzzle.
> >> >
> >> >
> >> > As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, t=
hat
> >> is -
> >> > a hypercube consisting of exploded skewbs (with additional cuts clip=
ping
> >> off
> >> > the corners). Now identify all the pieces affected by one particular
> >> > rotation and try to identify the move that is on the opposite side o=
f
> >> the
> >> > puzzle. Identified correctly, this opposite move does not affect any=
of
> >> the
> >> > same pieces. However, not every piece is affected by these two moves=
!
> >> There
> >> > is a band of pieces remaining untouched, much like the slice of a 3x=
3x3
> >> left
> >> > untouched by UD'. This means the puzzle is not deepcut! If we push t=
he
> >> 3D
> >> > hyper cutting planes deeper into the 4D puzzle, we get cells that lo=
ok
> >> like
> >> > Master Skewbs. Continuing to push, certain pieces of these Master Sk=
ewbs
> >> get
> >> > thinner and thinner until they vanish at the point when opposing
> >> hyperplanes
> >> > meet. This is the deepcut vertex turning 8Cell puzzle. Each cell loo=
ks
> >> like
> >> > an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cell=
s
> >> that
> >> > look like dino cubes that is shallower cut. Although these puzzles a=
re
> >> > visually identical, a single move on the shallower cut puzzle affect=
s
> >> pieces
> >> > on only 4 cells while a single move on the deepcut puzzle affects pi=
eces
> >> on
> >> > all 8 cells. Also of interest is the series of complicated looking
> >> puzzles
> >> > that appear at cut depths between the 4D Skewb and each of these din=
o
> >> cell
> >> > puzzles, although there are only 3 slices per axis in these puzzles
> >> (same
> >> > order as 3x3x3), each cell is an exploded Master Skewb!
> >> >
> >> > Although I have explored several other ideas, the three puzzles (4D =
FTO,
> >> 4D
> >> > Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
> >> > candidates for the next run of 4D puzzles, they implement some compl=
ex
> >> piece
> >> > interactions without becoming too large or too visually crowded.
> >> >
> >> > These puzzles are of an incredible interest to me, because the
> >> interactions
> >> > of the pieces are so much more intricate than the 120Cell or any of =
the
> >> > simplex vertex puzzles possible in the current MC4D program! As I
> >> mentioned
> >> > in my message to Roice, I have a good idea of how each of these puzz=
les
> >> look
> >> > and function and would gladly assist anyone (Roice? haha) who wants =
to
> >> > attempt to program it. In the meantime, I will take a look at the co=
de
> >> Roice
> >> > has provided me and try to do some work myself, but I highly doubt I
> >> will
> >> > have success without an experienced programmer's help ;)
> >> >
> >> > I would love to hear others' thoughts on these!
> >> > -Matt Galla
> >> >
> >>
> >>
> >
> >
> >
>
From: "Andrey" <andreyastrelin@yahoo.com>
Date: Wed, 26 Jan 2011 07:06:00 -0000
Subject: [MC4D] Re: Other 4D puzzles
So shallow-cut 24-cell will have 57 stickers in one cell, and 16-cell will =
have 25 stickers (4 corners, 4 sub-corners, 6 edges, 6 sub-edges, 4 faces a=
nd 1 center). I'm not sure if sub-corners and sub-edges will be 1C stickers=
or parts of some multisticker piece. In the first case we may want to have=
more complicated painting than "one color per face" to make it "super-16-c=
ell" :)
Hypercube with diagonal cuts may be the simplest puzzle (by structure, no=
t by implementation of difficulty). It's almost equivalent to two-layer 16-=
cell with cell-centered cutting, right? (I prefer "cell-, face-, edge- and =
vertex-centered" term. Or better, "3D-, 2D-, 1D- and 0D-centered" - we can'=
t use terms like 4C in these puzzles).
I wonder, what will be structure of cuts of shallow cell-centered cutting=
of 600-cell. Looks like ther will be 4 different types of vertex-incidence=
-induced cuttings, and one tetrahedron will be crossed by 56 planes. _That_=
will be difficult.=20
Andrey
From: Melinda Green <melinda@superliminal.com>
Date: Wed, 26 Jan 2011 01:04:50 -0800
Subject: Re: [MC4D] Re: Other 4D puzzles
Thanks for the links, Nan! I've ordered one from the first link along
with a floppy cube since that is the closest thing to MC2D. :-)
Oh, and a slightly sarcastic "thanks" to Andrey for mentioning a puzzle
based on the 600 cell. I've sort of thought of that monster as "The Name
We Must Not Say Aloud". Now that the spell is broken I suppose that at
some point someone is going to implement one and someone else will solve
it. I shudder to imagine how many hundreds of hours that solution will
require. OTOH, maybe since it is the duel of the 120 cell, it will be
almost exactly as hard. Predictions anyone?
-Melinda
On 1/25/2011 9:55 PM, schuma wrote:
> Hi,
>
> This message is dedicated to answer Roice's question about where to buy a face turning octahedron. Here are some suggestions:
>
> http://www.hknowstore.com/item.aspx?corpname=nowstore&itemid=058fbd01-4a44-4e6f-99ec-71ae3bd9eb23
>
> http://www.witeden.com/goods.php?id=174
>
> or ebay sellers, for example
>
> http://cgi.ebay.com/Magic-Octahedron-Rubiks-Cube-Star-Puzzler-Transparent-/160515420233?pt=LH_DefaultDomain_2&hash=item255f76f049
>
> This puzzle has been mass-produced twice, so they are pretty cheap now, for around $10+shipping. Shipping might take two weeks because the sellers are usually in Hong Kong or China. Just make sure it's a face-turning one before you buy it, because the vertex turning octahedron is also mass produced, which looks almost the same.
>
> Nan
>
> --- In 4D_Cubing@yahoogroups.com, Roice Nelson
>> Great stuff guys!
>>
>> Special thanks for helping me to picture the nature of a cell-turning
>> 24-cell puzzle. In trying to understand the extra cuts you described, I see
>> now that they are somewhat related to the
>> incidence
>> of adjacent cells. In particular, the unusual cuts come from
>> the adjacent cells with vertex-only incidences. (btw, "parallel to vertex"
>> rather than "perpendicular to vertex" seems like decent language, though
>> this wording does refer to the adjacent cell the cut is based on.) At first
>> I thought the 24-cell puzzle would also need cuts parallel to edges, but
>> there are no adjacent cells having incident edges which do not also
>> have incident planes. On the 16-cell, there are adjacent cells with all
>> three possible incidence types, and it looks like there will be three styles
>> of cuts on its tetrahedral cells. Both puzzles sound difficult! We've
>> never run into these kinds of situations before because the adjacent cells
>> on puzzles with simplex vertex figures all have incident planes.
>>
>> I also thought I'd mention that I never felt fully comfortable calling
>> Magic120Cell a "4D Megaminx", due to some of the analogy ambiguities you are
>> discussing. Similarly, my personal preference leans towards not using terms
>> like "4D Skewb", unless all could agree on the most defining Skewb-like
>> properties. Since a 3D Skewb is a vertex-turning puzzle with slices halfway
>> between diametrically opposed vertices, it could be argued that the 4D Skewb
>> must have all these properties, with the only change being that the
>> properties are now applied to a hypercube (in other words, that the 4D Skewb
>> is the puzzle you described that has faces that look like Dino cubes). I
>> guess my point is that I prefer language like Nan used, explicitly
>> describing the polytope and the nature of the twisting. But I also agree
>> the naming is not the most important aspect (and I've never been good at
>> creating interesting puzzle names), so that's all I will have to say about
>> that :)
>>
>> Cheers,
>> Roice
>>
>> P.S. Anyone know where you can buy the face-turning-octahedron puzzle? I'd
>> like to own one.
>>
>>
>> On Sun, Jan 23, 2011 at 2:54 PM, Galla, Matthew
>>
>>>
>>> Ah,
>>>
>>> schuma (Nan?) is quite right. A very "natural" (perhaps even the most
>>> "natural") extension of the FTO to 4D is a cell turning 16Cell. However, I
>>> was looking for a puzzle where each cell looks like an FTO, and this
>>> obviously cannot be the case for a 16Cell, which has tetrahedral faces.
>>>
>>> It seems that in 4D there are two ways of interpreting the analogue of some
>>> puzzles.
>>> On the one hand, you could construct a 4D puzzle where every cell looks
>>> like the 3D counterpart, in the case of all puzzles except tetrahedral and
>>> icosahedral, this unambiguously assigns the 4D shape. In the case of a
>>> tetrahedral puzzle, you can choose between the 5Cell, the 16Cell, and the
>>> 600Cell. In the case of icosahedral, this interpretation fails to produce an
>>> equivalent puzzle.
>>> On the other hand, you can analyze the construction of the 3D shape and
>>> construct the equivalent 4D shape. In the case of the octahedron, 4
>>> triangles meet at a point (triangle being the 2-simplex). Thus the 4D
>>> equivalent should have 4 tetrahedra (tetrahedron being the
>>> 3-simplex) meeting at an edge. This can unambiguously find analogues for all
>>> regular polyhedra (in the case of the FTO, this interpretation gives a
>>> 16Cell with pyraminx-like cells, the one schuma is referring to), and
>>> possibly more; however no puzzle will ever get mapped to the 24Cell (because
>>> the 24Cell has no 3D equivalent).
>>>
>>> I realize the first method given above is "artificial" in a sense. You do
>>> not design a 3D puzzle by first deciding what each face should look like and
>>> then repeating it over the rest of the puzzle. BUT YOU COULD! ;) As long as
>>> you pick a face that is cut in such a way that all cuts are parallel to the
>>> sides of the face-shape and at equal depths, the resulting puzzle should be
>>> "playable". (the 4D analogue for this is choosing a cell layout such that
>>> all cuts are parallel to the faces of the cell and at equal depths - but
>>> this is PRECISELY what allows the cell to alone be a 3D puzzle)
>>>
>>> In any case, it seems that both methods produce valid puzzles, and while
>>> some 4D puzzles can be obtained through either interpretation, there are
>>> some (like the 24Cell 4D FTO I described earlier) that can only be produced
>>> through one interpretation. I therefore think it is important that we
>>> consider both interpretations (plus I think a 24Cell would be more exciting,
>>> but maybe that's just me ;) )
>>>
>>>
>>> Thanks for bringing that up schuma!
>>>
>>> -Matt Galla
>>>
>>> PS On TP my username is Allagem ;)
>>>
>>> On Sun, Jan 23, 2011 at 12:44 PM, schuma
>>>
>>>>
>>>> Hi Matt,
>>>>
>>>> Thank you for starting the discussions about other 4D puzzles.
>>>>
>>>> Can you explain more about why the 4D analogue of the FTO is a 24-cell
>>>> instead of a 16-cell? Although the faces of the 24-cell are octahedra,
>>>> 24-cell is a self-dual polytope that is not a simplex. From this point of
>>>> view, it has no 3D analog. In fact it has no analog in any dimension other
>>>> than 4D. However, the 16-cell belongs to the family of cross-polytopes,
>>>> which are the duals of hypercubes, and exist in any number of dimensions. (
>>>> http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polytope
>>>> is 16-cell. Therefore I think a natural extension of FTO is a cell-turning
>>>> 16-cell, because they share more similarities.
>>>>
>>>> For example, you may know that in 3D, the FTO can be regarded as a
>>>> shape-mod of Rex Cube, a vertex turning cube (
>>>> http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=12659). If the 4D
>>>> FTO is a shape-mod of the vertex turning hypercube, it should be a
>>>> cell-turning 16-cell instead of a cell-turning 24-cell.
>>>>
>>>> No matter calling it 4D FTO or else, I believe what you have described in
>>>> the third paragraph is a cell-turning 24-cell. It should be an amazing
>>>> puzzle to solve. I have special feeling about it because of its uniqueness
>>>> in all the dimensions.
>>>>
>>>> Nan
>>>>
>>>>
>>>> --- In 4D_Cubing@yahoogroups.com<4D_Cubing%40yahoogroups.com>, "Galla,
>>>> Matthew"
>>>>> Hey everyone,
>>>>>
>>>>> As I mentioned in my response about my solve of the 120Cell, I have been
>>>>> looking into some other 4D puzzles and have worked out how several of
>>>> these
>>>>> puzzles should work and even discovered some interesting properties.
>>>> Here is
>>>>> a snipet from my 120Cell solve message I sent Roice discussing this
>>>> subject:
>>>>> "I am still hoping for more complicated 4D puzzles and am willing to do
>>>>> whatever I can to help make them a reality. Coding a 4d space like you
>>>> have
>>>>> is quite intimidating, but perhaps I can try to build off a pre-existing
>>>> one
>>>>> with some guidance. I have already determined what the 4D analogue of
>>>> the
>>>>> FTO (face turning octahedron, invented some time last year if you have
>>>> not
>>>>> already seen it) would look like and how it would function as well as
>>>> the 4D
>>>>> analogue of the Skewb and Helicopter Cube (on that note I also have a
>>>>> suggestion as to how to make the interface for 4D puzzles that are
>>>> non-face
>>>>> rotating, like the Skewb and Helicopter Cube). I have also made some
>>>>> interesting discoveries like for example making a 4D puzzle out of a 3D
>>>>> puzzle can make some additional internal cuts without altering the
>>>> exterior
>>>>> of a 3D face (true for all three puzzle I mentioned so far) and how a 4D
>>>>> Skewb is not deepcut! (that is every cell looks like a Skewb and seems
>>>> to
>>>>> behave as such) The vertex turning deepcut hypercube has faces that
>>>>> externally each look like a dino cube. Is there anything I can do to
>>>> make
>>>>> help make these a reality? After spending 150 hours on the 120Cell, I
>>>> can
>>>>> honestly say that about 146 of the hours all feel exactly the same and I
>>>> am
>>>>> dying to find a more interesting 4D puzzle to explore :)"
>>>>>
>>>>> To expand a little on some of the things I mentioned above, the 4D FTO
>>>> would
>>>>> be a 24Cell with faces that look like an exploded version of this
>>>> puzzle:
>>>>> http://www.jaapsch.net/puzzles/octaface.htm
>>>>> with one big difference, in addition to every cut on the 3D analogue of
>>>> the
>>>>> puzzle, the 4D version has and additional cut perpendicular to the
>>>> vertices
>>>>> of each face that line up with first cut down. :/ Sorry, I know that
>>>> wasn't
>>>>> very well worded and I'm not sure how well sending a picture would work
>>>>> through a yahoo group. Let me try again: these extra cuts would
>>>> essential
>>>>> cut off the vertex pieces of each cell. Removing the pieces that are
>>>>> affected by this new unexpected cut will result in cells that have an
>>>>> exterior that matches this puzzle:
>>>>> http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=451
>>>>> (If you can follow my inadequate descriptions above, the 4D FTO would
>>>> have 6
>>>>> distinct visible pieces, not just the 5 present on an exploded 3D FTO -
>>>> the
>>>>> extra comes from splitting each of the vertex pieces of the 3D Fto in
>>>> half)
>>>>> A similar phenomenon occurs on both the 4D helicopter cube (3D:
>>>>>
>>>> http://www.puzzleforge.com/main/index.php?option=com_content&view=article&id=49:hcannounce&catid=1:latest-news&Itemid=50
>>>> )
>>>>> and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by
>>>> analogue, I
>>>>> mean each cell looks like the respective puzzle and moves in a similar
>>>>> manner]. In each of these puzzles, the new cut clips off the corners.
>>>>> Remembering that to truly express the 4D nature of these puzzles, each
>>>> cell
>>>>> must be "exploded", so what used to be he vertex pieces for each of
>>>> these
>>>>> puzzles have now been cut in half resulting in an internal piece that
>>>>> behaves as one might have expected the single original piece to act and
>>>> an
>>>>> external piece that in addition to moving every time the internal piece
>>>>> moves, can also be affected by a non-adjacent face.
>>>>>
>>>>>
>>>>> As to a nice interface for non-face rotating 4D puzzles, my suggestion
>>>> is to
>>>>> display the wireframe of a 3D solid that displays all the symmetries
>>>> implied
>>>>> by the rotation between the faces and perform clicks not on the puzzle
>>>>> itself, but only on this wireframe. For example, on a 4D Skewb,
>>>> rotations
>>>>> are made around the "corners" of each cell. These rotations are all
>>>>> equivalent to some rotation on a face turning 16Cell. So, in the
>>>> Hypercube
>>>>> shape, we could display wireframes of tetrahedrons that "float" between
>>>> the
>>>>> appropriate corners of 4 hypercube cells. When the user clicks on a face
>>>> of
>>>>> this floating wirefram tetrahedron, both the tetrahedron and the pieces
>>>>> affected by the corresponding "vertex twist" all rotate. Clicking on the
>>>>> actual stickers of the puzzle does nothing; all rotations are executed
>>>> by
>>>>> clicking on these "rotation polyhedra". In the case of the 4D Helicopter
>>>>> Cube, the appropriate wireframe shape would be a triangular prism -
>>>>> rotations around both the triangle faces and the rectangular faces are
>>>>> possible moves on the 4D Helicopter Cube, and each of these rotations
>>>> can be
>>>>> executed unambiguously by clicking on the appropriate face of the
>>>> triangular
>>>>> prism wireframe floating between the cells of the puzzle.
>>>>>
>>>>>
>>>>> As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, that
>>>> is -
>>>>> a hypercube consisting of exploded skewbs (with additional cuts clipping
>>>> off
>>>>> the corners). Now identify all the pieces affected by one particular
>>>>> rotation and try to identify the move that is on the opposite side of
>>>> the
>>>>> puzzle. Identified correctly, this opposite move does not affect any of
>>>> the
>>>>> same pieces. However, not every piece is affected by these two moves!
>>>> There
>>>>> is a band of pieces remaining untouched, much like the slice of a 3x3x3
>>>> left
>>>>> untouched by UD'. This means the puzzle is not deepcut! If we push the
>>>> 3D
>>>>> hyper cutting planes deeper into the 4D puzzle, we get cells that look
>>>> like
>>>>> Master Skewbs. Continuing to push, certain pieces of these Master Skewbs
>>>> get
>>>>> thinner and thinner until they vanish at the point when opposing
>>>> hyperplanes
>>>>> meet. This is the deepcut vertex turning 8Cell puzzle. Each cell looks
>>>> like
>>>>> an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells
>>>> that
>>>>> look like dino cubes that is shallower cut. Although these puzzles are
>>>>> visually identical, a single move on the shallower cut puzzle affects
>>>> pieces
>>>>> on only 4 cells while a single move on the deepcut puzzle affects pieces
>>>> on
>>>>> all 8 cells. Also of interest is the series of complicated looking
>>>> puzzles
>>>>> that appear at cut depths between the 4D Skewb and each of these dino
>>>> cell
>>>>> puzzles, although there are only 3 slices per axis in these puzzles
>>>> (same
>>>>> order as 3x3x3), each cell is an exploded Master Skewb!
>>>>>
>>>>> Although I have explored several other ideas, the three puzzles (4D FTO,
>>>> 4D
>>>>> Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
>>>>> candidates for the next run of 4D puzzles, they implement some complex
>>>> piece
>>>>> interactions without becoming too large or too visually crowded.
>>>>>
>>>>> These puzzles are of an incredible interest to me, because the
>>>> interactions
>>>>> of the pieces are so much more intricate than the 120Cell or any of the
>>>>> simplex vertex puzzles possible in the current MC4D program! As I
>>>> mentioned
>>>>> in my message to Roice, I have a good idea of how each of these puzzles
>>>> look
>>>>> and function and would gladly assist anyone (Roice? haha) who wants to
>>>>> attempt to program it. In the meantime, I will take a look at the code
>>>> Roice
>>>>> has provided me and try to do some work myself, but I highly doubt I
>>>> will
>>>>> have success without an experienced programmer's help ;)
>>>>>
>>>>> I would love to hear others' thoughts on these!
>>>>> -Matt Galla
>>>>>
>>>>
>>>
>>>
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>
From: "schuma" <mananself@gmail.com>
Date: Wed, 26 Jan 2011 10:19:37 -0000
Subject: [MC4D] Re: Other 4D puzzles
Hi guys,
I guess the shallow-cut 600-cell is MUCH MUCH harder than the 120-cell rath=
er than "almost exactly as hard". The reason is that there are too many sma=
ll pieces due to "incidence" (I don't know the exact meaning of this term, =
but I do have some kind of intuition).=20
My guess comes from the experience in 3D. We all know the neat structure of=
a megaminx. But what about the shallow-cut icosahedron? Does it have a nea=
t shape? It looks like this:=20
http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/icosa_f3.gif
Even using the shallowest cuts, we inevitably have many small pieces. It's =
because around each vertex, five cutting planes intersect each other to cre=
ate them. I guess this is the vertex-incidence properties that Roice and An=
drey talked about. As a result, solving such a puzzle is much harder than s=
olving a megaminx. The number of steps to solve it is usually an order of m=
agnitude more than that for a megaminx.
I think the 600-cell puzzle has a similar issue. At each vertex there is a =
12C pieces. And around it, 12 hyperplanes intersect, producing numerous sma=
ll pieces. The number of pieces in a shallow-cut 600-cell must be several t=
imes more than that of 120-cell. It's just horrible.=20
Anyway to make it better?=20
(1) Simply drop some small pieces.=20
(2) Make the cuts curvy to avoid intersection. 3D Examples for icosahedron:=
=20
http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/icosa_f9.gif
and physical puzzle:
http://www.puzzleforge.com/images/photos/twistypuzzlesposts/radiolarian/sti=
ckered/Picture1%20079.jpg
(3) Make a vertex turning 120-cell rather than cell-turning 600-cell. Altho=
ugh it sounds like nothing is changed, it actually makes things much easier=
. The shallowest vertex turning 120-cell only contains trivial tips. If we =
slowly make the cuts deeper and deeper, we are slowly introducing new types=
of pieces. We can always find a depth that produces the right number of pi=
eces.
-- Nan
--- In 4D_Cubing@yahoogroups.com, Melinda Green
>
> Thanks for the links, Nan! I've ordered one from the first link along=20
> with a floppy cube since that is the closest thing to MC2D. :-)
>=20
> Oh, and a slightly sarcastic "thanks" to Andrey for mentioning a puzzle=20
> based on the 600 cell. I've sort of thought of that monster as "The Name=
=20
> We Must Not Say Aloud". Now that the spell is broken I suppose that at=20
> some point someone is going to implement one and someone else will solve=
=20
> it. I shudder to imagine how many hundreds of hours that solution will=20
> require. OTOH, maybe since it is the duel of the 120 cell, it will be=20
> almost exactly as hard. Predictions anyone?
>=20
> -Melinda
>=20
> On 1/25/2011 9:55 PM, schuma wrote:
> > Hi,
> >
> > This message is dedicated to answer Roice's question about where to buy=
a face turning octahedron. Here are some suggestions:
> >
> > http://www.hknowstore.com/item.aspx?corpname=3Dnowstore&itemid=3D058fbd=
01-4a44-4e6f-99ec-71ae3bd9eb23
> >
> > http://www.witeden.com/goods.php?id=3D174
> >
> > or ebay sellers, for example
> >
> > http://cgi.ebay.com/Magic-Octahedron-Rubiks-Cube-Star-Puzzler-Transpare=
nt-/160515420233?pt=3DLH_DefaultDomain_2&hash=3Ditem255f76f049
> >
> > This puzzle has been mass-produced twice, so they are pretty cheap now,=
for around $10+shipping. Shipping might take two weeks because the sellers=
are usually in Hong Kong or China. Just make sure it's a face-turning one =
before you buy it, because the vertex turning octahedron is also mass produ=
ced, which looks almost the same.
> >
> > Nan
> >
> > --- In 4D_Cubing@yahoogroups.com, Roice Nelson
> >> Great stuff guys!
> >>
> >> Special thanks for helping me to picture the nature of a cell-turning
> >> 24-cell puzzle. In trying to understand the extra cuts you described,=
I see
> >> now that they are somewhat related to the
> >> incidence
> >> of adjacent cells. In particular, the unusual cuts come from
> >> the adjacent cells with vertex-only incidences. (btw, "parallel to ver=
tex"
> >> rather than "perpendicular to vertex" seems like decent language, thou=
gh
> >> this wording does refer to the adjacent cell the cut is based on.) At=
first
> >> I thought the 24-cell puzzle would also need cuts parallel to edges, b=
ut
> >> there are no adjacent cells having incident edges which do not also
> >> have incident planes. On the 16-cell, there are adjacent cells with a=
ll
> >> three possible incidence types, and it looks like there will be three =
styles
> >> of cuts on its tetrahedral cells. Both puzzles sound difficult! We'v=
e
> >> never run into these kinds of situations before because the adjacent c=
ells
> >> on puzzles with simplex vertex figures all have incident planes.
> >>
> >> I also thought I'd mention that I never felt fully comfortable calling
> >> Magic120Cell a "4D Megaminx", due to some of the analogy ambiguities y=
ou are
> >> discussing. Similarly, my personal preference leans towards not using=
terms
> >> like "4D Skewb", unless all could agree on the most defining Skewb-lik=
e
> >> properties. Since a 3D Skewb is a vertex-turning puzzle with slices h=
alfway
> >> between diametrically opposed vertices, it could be argued that the 4D=
Skewb
> >> must have all these properties, with the only change being that the
> >> properties are now applied to a hypercube (in other words, that the 4D=
Skewb
> >> is the puzzle you described that has faces that look like Dino cubes).=
I
> >> guess my point is that I prefer language like Nan used, explicitly
> >> describing the polytope and the nature of the twisting. But I also ag=
ree
> >> the naming is not the most important aspect (and I've never been good =
at
> >> creating interesting puzzle names), so that's all I will have to say a=
bout
> >> that :)
> >>
> >> Cheers,
> >> Roice
> >>
> >> P.S. Anyone know where you can buy the face-turning-octahedron puzzle?=
I'd
> >> like to own one.
> >>
> >>
> >> On Sun, Jan 23, 2011 at 2:54 PM, Galla, Matthew
> >>
> >>>
> >>> Ah,
> >>>
> >>> schuma (Nan?) is quite right. A very "natural" (perhaps even the most
> >>> "natural") extension of the FTO to 4D is a cell turning 16Cell. Howev=
er, I
> >>> was looking for a puzzle where each cell looks like an FTO, and this
> >>> obviously cannot be the case for a 16Cell, which has tetrahedral face=
s.
> >>>
> >>> It seems that in 4D there are two ways of interpreting the analogue o=
f some
> >>> puzzles.
> >>> On the one hand, you could construct a 4D puzzle where every cell loo=
ks
> >>> like the 3D counterpart, in the case of all puzzles except tetrahedra=
l and
> >>> icosahedral, this unambiguously assigns the 4D shape. In the case of =
a
> >>> tetrahedral puzzle, you can choose between the 5Cell, the 16Cell, and=
the
> >>> 600Cell. In the case of icosahedral, this interpretation fails to pro=
duce an
> >>> equivalent puzzle.
> >>> On the other hand, you can analyze the construction of the 3D shape a=
nd
> >>> construct the equivalent 4D shape. In the case of the octahedron, 4
> >>> triangles meet at a point (triangle being the 2-simplex). Thus the 4D
> >>> equivalent should have 4 tetrahedra (tetrahedron being the
> >>> 3-simplex) meeting at an edge. This can unambiguously find analogues =
for all
> >>> regular polyhedra (in the case of the FTO, this interpretation gives =
a
> >>> 16Cell with pyraminx-like cells, the one schuma is referring to), and
> >>> possibly more; however no puzzle will ever get mapped to the 24Cell (=
because
> >>> the 24Cell has no 3D equivalent).
> >>>
> >>> I realize the first method given above is "artificial" in a sense. Yo=
u do
> >>> not design a 3D puzzle by first deciding what each face should look l=
ike and
> >>> then repeating it over the rest of the puzzle. BUT YOU COULD! ;) As l=
ong as
> >>> you pick a face that is cut in such a way that all cuts are parallel =
to the
> >>> sides of the face-shape and at equal depths, the resulting puzzle sho=
uld be
> >>> "playable". (the 4D analogue for this is choosing a cell layout such =
that
> >>> all cuts are parallel to the faces of the cell and at equal depths - =
but
> >>> this is PRECISELY what allows the cell to alone be a 3D puzzle)
> >>>
> >>> In any case, it seems that both methods produce valid puzzles, and wh=
ile
> >>> some 4D puzzles can be obtained through either interpretation, there =
are
> >>> some (like the 24Cell 4D FTO I described earlier) that can only be pr=
oduced
> >>> through one interpretation. I therefore think it is important that we
> >>> consider both interpretations (plus I think a 24Cell would be more ex=
citing,
> >>> but maybe that's just me ;) )
> >>>
> >>>
> >>> Thanks for bringing that up schuma!
> >>>
> >>> -Matt Galla
> >>>
> >>> PS On TP my username is Allagem ;)
> >>>
> >>> On Sun, Jan 23, 2011 at 12:44 PM, schuma
> >>>
> >>>>
> >>>> Hi Matt,
> >>>>
> >>>> Thank you for starting the discussions about other 4D puzzles.
> >>>>
> >>>> Can you explain more about why the 4D analogue of the FTO is a 24-ce=
ll
> >>>> instead of a 16-cell? Although the faces of the 24-cell are octahedr=
a,
> >>>> 24-cell is a self-dual polytope that is not a simplex. From this poi=
nt of
> >>>> view, it has no 3D analog. In fact it has no analog in any dimension=
other
> >>>> than 4D. However, the 16-cell belongs to the family of cross-polytop=
es,
> >>>> which are the duals of hypercubes, and exist in any number of dimens=
ions. (
> >>>> http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polyt=
ope
> >>>> is 16-cell. Therefore I think a natural extension of FTO is a cell-t=
urning
> >>>> 16-cell, because they share more similarities.
> >>>>
> >>>> For example, you may know that in 3D, the FTO can be regarded as a
> >>>> shape-mod of Rex Cube, a vertex turning cube (
> >>>> http://www.twistypuzzles.com/forum/viewtopic.php?f=3D15&t=3D12659). =
If the 4D
> >>>> FTO is a shape-mod of the vertex turning hypercube, it should be a
> >>>> cell-turning 16-cell instead of a cell-turning 24-cell.
> >>>>
> >>>> No matter calling it 4D FTO or else, I believe what you have describ=
ed in
> >>>> the third paragraph is a cell-turning 24-cell. It should be an amazi=
ng
> >>>> puzzle to solve. I have special feeling about it because of its uniq=
ueness
> >>>> in all the dimensions.
> >>>>
> >>>> Nan
> >>>>
> >>>>
> >>>> --- In 4D_Cubing@yahoogroups.com<4D_Cubing%40yahoogroups.com>, "Gall=
a,
> >>>> Matthew"
> >>>>> Hey everyone,
> >>>>>
> >>>>> As I mentioned in my response about my solve of the 120Cell, I have=
been
> >>>>> looking into some other 4D puzzles and have worked out how several =
of
> >>>> these
> >>>>> puzzles should work and even discovered some interesting properties=
.
> >>>> Here is
> >>>>> a snipet from my 120Cell solve message I sent Roice discussing this
> >>>> subject:
> >>>>> "I am still hoping for more complicated 4D puzzles and am willing t=
o do
> >>>>> whatever I can to help make them a reality. Coding a 4d space like =
you
> >>>> have
> >>>>> is quite intimidating, but perhaps I can try to build off a pre-exi=
sting
> >>>> one
> >>>>> with some guidance. I have already determined what the 4D analogue =
of
> >>>> the
> >>>>> FTO (face turning octahedron, invented some time last year if you h=
ave
> >>>> not
> >>>>> already seen it) would look like and how it would function as well =
as
> >>>> the 4D
> >>>>> analogue of the Skewb and Helicopter Cube (on that note I also have=
a
> >>>>> suggestion as to how to make the interface for 4D puzzles that are
> >>>> non-face
> >>>>> rotating, like the Skewb and Helicopter Cube). I have also made som=
e
> >>>>> interesting discoveries like for example making a 4D puzzle out of =
a 3D
> >>>>> puzzle can make some additional internal cuts without altering the
> >>>> exterior
> >>>>> of a 3D face (true for all three puzzle I mentioned so far) and how=
a 4D
> >>>>> Skewb is not deepcut! (that is every cell looks like a Skewb and se=
ems
> >>>> to
> >>>>> behave as such) The vertex turning deepcut hypercube has faces that
> >>>>> externally each look like a dino cube. Is there anything I can do t=
o
> >>>> make
> >>>>> help make these a reality? After spending 150 hours on the 120Cell,=
I
> >>>> can
> >>>>> honestly say that about 146 of the hours all feel exactly the same =
and I
> >>>> am
> >>>>> dying to find a more interesting 4D puzzle to explore :)"
> >>>>>
> >>>>> To expand a little on some of the things I mentioned above, the 4D =
FTO
> >>>> would
> >>>>> be a 24Cell with faces that look like an exploded version of this
> >>>> puzzle:
> >>>>> http://www.jaapsch.net/puzzles/octaface.htm
> >>>>> with one big difference, in addition to every cut on the 3D analogu=
e of
> >>>> the
> >>>>> puzzle, the 4D version has and additional cut perpendicular to the
> >>>> vertices
> >>>>> of each face that line up with first cut down. :/ Sorry, I know tha=
t
> >>>> wasn't
> >>>>> very well worded and I'm not sure how well sending a picture would =
work
> >>>>> through a yahoo group. Let me try again: these extra cuts would
> >>>> essential
> >>>>> cut off the vertex pieces of each cell. Removing the pieces that ar=
e
> >>>>> affected by this new unexpected cut will result in cells that have =
an
> >>>>> exterior that matches this puzzle:
> >>>>> http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=3D451
> >>>>> (If you can follow my inadequate descriptions above, the 4D FTO wou=
ld
> >>>> have 6
> >>>>> distinct visible pieces, not just the 5 present on an exploded 3D F=
TO -
> >>>> the
> >>>>> extra comes from splitting each of the vertex pieces of the 3D Fto =
in
> >>>> half)
> >>>>> A similar phenomenon occurs on both the 4D helicopter cube (3D:
> >>>>>
> >>>> http://www.puzzleforge.com/main/index.php?option=3Dcom_content&view=
=3Darticle&id=3D49:hcannounce&catid=3D1:latest-news&Itemid=3D50
> >>>> )
> >>>>> and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by
> >>>> analogue, I
> >>>>> mean each cell looks like the respective puzzle and moves in a simi=
lar
> >>>>> manner]. In each of these puzzles, the new cut clips off the corner=
s.
> >>>>> Remembering that to truly express the 4D nature of these puzzles, e=
ach
> >>>> cell
> >>>>> must be "exploded", so what used to be he vertex pieces for each of
> >>>> these
> >>>>> puzzles have now been cut in half resulting in an internal piece th=
at
> >>>>> behaves as one might have expected the single original piece to act=
and
> >>>> an
> >>>>> external piece that in addition to moving every time the internal p=
iece
> >>>>> moves, can also be affected by a non-adjacent face.
> >>>>>
> >>>>>
> >>>>> As to a nice interface for non-face rotating 4D puzzles, my suggest=
ion
> >>>> is to
> >>>>> display the wireframe of a 3D solid that displays all the symmetrie=
s
> >>>> implied
> >>>>> by the rotation between the faces and perform clicks not on the puz=
zle
> >>>>> itself, but only on this wireframe. For example, on a 4D Skewb,
> >>>> rotations
> >>>>> are made around the "corners" of each cell. These rotations are all
> >>>>> equivalent to some rotation on a face turning 16Cell. So, in the
> >>>> Hypercube
> >>>>> shape, we could display wireframes of tetrahedrons that "float" bet=
ween
> >>>> the
> >>>>> appropriate corners of 4 hypercube cells. When the user clicks on a=
face
> >>>> of
> >>>>> this floating wirefram tetrahedron, both the tetrahedron and the pi=
eces
> >>>>> affected by the corresponding "vertex twist" all rotate. Clicking o=
n the
> >>>>> actual stickers of the puzzle does nothing; all rotations are execu=
ted
> >>>> by
> >>>>> clicking on these "rotation polyhedra". In the case of the 4D Helic=
opter
> >>>>> Cube, the appropriate wireframe shape would be a triangular prism -
> >>>>> rotations around both the triangle faces and the rectangular faces =
are
> >>>>> possible moves on the 4D Helicopter Cube, and each of these rotatio=
ns
> >>>> can be
> >>>>> executed unambiguously by clicking on the appropriate face of the
> >>>> triangular
> >>>>> prism wireframe floating between the cells of the puzzle.
> >>>>>
> >>>>>
> >>>>> As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, =
that
> >>>> is -
> >>>>> a hypercube consisting of exploded skewbs (with additional cuts cli=
pping
> >>>> off
> >>>>> the corners). Now identify all the pieces affected by one particula=
r
> >>>>> rotation and try to identify the move that is on the opposite side =
of
> >>>> the
> >>>>> puzzle. Identified correctly, this opposite move does not affect an=
y of
> >>>> the
> >>>>> same pieces. However, not every piece is affected by these two move=
s!
> >>>> There
> >>>>> is a band of pieces remaining untouched, much like the slice of a 3=
x3x3
> >>>> left
> >>>>> untouched by UD'. This means the puzzle is not deepcut! If we push =
the
> >>>> 3D
> >>>>> hyper cutting planes deeper into the 4D puzzle, we get cells that l=
ook
> >>>> like
> >>>>> Master Skewbs. Continuing to push, certain pieces of these Master S=
kewbs
> >>>> get
> >>>>> thinner and thinner until they vanish at the point when opposing
> >>>> hyperplanes
> >>>>> meet. This is the deepcut vertex turning 8Cell puzzle. Each cell lo=
oks
> >>>> like
> >>>>> an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cel=
ls
> >>>> that
> >>>>> look like dino cubes that is shallower cut. Although these puzzles =
are
> >>>>> visually identical, a single move on the shallower cut puzzle affec=
ts
> >>>> pieces
> >>>>> on only 4 cells while a single move on the deepcut puzzle affects p=
ieces
> >>>> on
> >>>>> all 8 cells. Also of interest is the series of complicated looking
> >>>> puzzles
> >>>>> that appear at cut depths between the 4D Skewb and each of these di=
no
> >>>> cell
> >>>>> puzzles, although there are only 3 slices per axis in these puzzles
> >>>> (same
> >>>>> order as 3x3x3), each cell is an exploded Master Skewb!
> >>>>>
> >>>>> Although I have explored several other ideas, the three puzzles (4D=
FTO,
> >>>> 4D
> >>>>> Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
> >>>>> candidates for the next run of 4D puzzles, they implement some comp=
lex
> >>>> piece
> >>>>> interactions without becoming too large or too visually crowded.
> >>>>>
> >>>>> These puzzles are of an incredible interest to me, because the
> >>>> interactions
> >>>>> of the pieces are so much more intricate than the 120Cell or any of=
the
> >>>>> simplex vertex puzzles possible in the current MC4D program! As I
> >>>> mentioned
> >>>>> in my message to Roice, I have a good idea of how each of these puz=
zles
> >>>> look
> >>>>> and function and would gladly assist anyone (Roice? haha) who wants=
to
> >>>>> attempt to program it. In the meantime, I will take a look at the c=
ode
> >>>> Roice
> >>>>> has provided me and try to do some work myself, but I highly doubt =
I
> >>>> will
> >>>>> have success without an experienced programmer's help ;)
> >>>>>
> >>>>> I would love to hear others' thoughts on these!
> >>>>> -Matt Galla
> >>>>>
> >>>>
> >>>
> >>>
> >
> >
> >
> > ------------------------------------
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
>
From: "Galla, Matthew" <mgalla@trinity.edu>
Date: Wed, 26 Jan 2011 04:52:00 -0600
Subject: Re: [MC4D] Re: Other 4D puzzles
--00151749f6a2622120049abd9f37
Content-Type: text/plain; charset=ISO-8859-1
Hey guys,
Thanks for really running with this idea! I hope you are starting to feel
some of the excitement I've had for a long time :)
Roice:
I think I will take your suggested terminology of parallel to vertex,
thanks! And yes, I think you are associating the extra cut to the right
phenomenon. Now there may be a more mathematical reason for this, but my
impression is that these extra cuts are caused by the number of cells
meeting at a edge and at a vertex in the 4D shape. In the case of the
24Cell, 3 cells meet at every edge (and 2 faces meet at every edge in an
octahedron). When one of these cells is rotated, it also moves one layer of
each of the other 2 cells, while the pieces "in common" of these two layers
are pairs of stickers that represent the same group of pieces. However, 6
cells meet at every vertex (and 4 faces meet at every vertex in an
octahedron). When one of these cells is rotated, it also moves one layer of
4 of the other cells (see the correlations in the numbers?). However, the
remaining cell is also affected and it is from this interaction the
additional cut is born, parallel to the vertex (and it must line up with the
cut depth of the cuts parallel to the faces of the cell). On the 16Cell, 4
cells meet at an edge, leaving one that receives a new cut parallel to the
edge; and 8 cells meet at a vertex: moving one moves 3 via face cuts, and 3
via edge cuts. The remaining 1 receives a new cut parallel to the vertex. So
yes, a cell-turning 16Cell exhibits all 3 cut types. I also agree with your
hesitation on the ambiguous names for these monsters. I was merely using 4D
Skewb and 24Cell FTO in lieu of a better name for discussion purposes.
Though I am having trouble seeing how 4D megaminx can possibly refer to
anything but the Magic120Cell.... Perhaps I need to ponder that longer.
Andrey:
Yes, you got the 24-cell I had in mind exactly right. There would be 57
pieces in one cell: 6 corners (6C), 6 sub-corners (1C), 12 edges (3C), 24
"offset faces" (2C), 8 "sub-faces" (!!!!!!!2C!!!!!!!), and 1 center (1C)
[grand total of 672 visible pieces for the whole puzzle, I believe]. For
this puzzle the sub-corners are not mathematically connected in any way, so
each one acts independently and would naturally have only one color.
However, what I am calling the "sub-faces" (attempting to use your
terminology) ARE mathematically connected in pairs (in fact I believe they
would be one physical piece if the 4D puzzle could be constructed in
physical form). Because of this, each piece has 2 colors (stickers),
making their orientation partially distinguishable: only 3 of the 6 possible
orientations would be accepted as correct. Adding extra colors to make the
orientation (and permutation in the case of the sub-corners!) of these
pieces visible is of course an option, just as it was on the 4D cube and
120Cell with the centers. As neither of those puzzles used them, I was sort
of leaning toward not adding "super" colors, but there is definitely
something missing without them. Maybe it could be optional to add "super
stickers"? :)
As far as the face-turning 16 cell goes, the number of pieces depends on
your cut depth. I have to admit, I was too interested in the 24Cell to
really consider any tetrahedral-faced puzzle but I certainly can now.
Assuming a cut shallow enough that no deeper interactions have occurred, I
believe you get a few more pieces than you have listed. Applying all 3 types
of cuts, I THINK I am seeing:
4 corners [rotated by 3 face cuts, 3 edge cuts, and 1 vertex cut] (8C),
4 sub-corners [rotated by 3 face cuts and 3 edge cuts] (1C),
12 "offset faces" [rotated by 3 face cuts and 2 edge cuts] (2C),
12 "offset sub-edges" [rotated by 3 face cuts and 1 edge cut] (1C), (these
are tricky - they are located between 2 "offset face" stickers and 1 edge
sticker)
4 "sub-sub-corners" [rotated by 3 face cuts] (1C) (Dunno what else to call
these - they are located exactly between a sub-corner sticker and a center
sticker)
6 edges [rotated by 2 face cuts and 1 edge cut] (4C),
6 sub-edges [rotated by 2 face cuts] (1C)
4 faces [rotated by 1 facecut] (2C)
1 center [rotated by nothing but cell rotation itself] (1C)
[it appears that rotation by x face cuts, y edge cuts, and z vertex cuts is
impossible if x
That gives a total of 53 stickers per cell! [for 592 pieces total in the
whole puzzle, I believe] That's almost as bad as the 24Cell ;) Also as near
as I can tell, increasing depth changes nothing until the center sticker is
squeezed out of existence (or, more accurately, the 3D surface of that
cell). After that, things get complicated....
And yes a hypercube with "Skewb-like" (supposedly what you mean by
diagonal?) cuts would be a 4D shape mod of a cell turning 16Cell *shudders
at thought of 4D shape-modding....* Depending on the exact nature of the
geometry (I haven't looked into it yet), a few pieces may be lost or gained
going from one puzzle to the other, but proper depths should preserve most
piece types between the two puzzles.
As for a 600Cell puzzle, I starting pursuing this in order to find more
interesting 4D puzzles without getting as big and repetitive as the 120Cell.
Don't you think the 600Cell is headed in the wrong direction? ;) The
symmetry guarantees there will be no more than 14400 of any chiral pieces
and no more than 7200 of any non-chiral pieces (the same guarantee we have
for cell-turning 120Cell puzzles), but still, the numbers can get ridiculous
really fast on a 600Cell :)
Also, for shallow-cut 600Cell puzzles, it appears you are right: there are 5
different cuts appearing. In addition to the standard face cut, there are a
pair of angled, mirrored cuts due to edge interactions (that can probably be
considered only one type of cut) and 4 types of cuts due to vertex
interactions (of which 2 are mirror images of each other and probably best
considered as only one type). This gives types of cuts that are not set
parallel to the faces of each cell in addition to the one type that is set
parallel to each face. Not only would this puzzle have an obscene number of
each type of piece. There will be dozens of different piece types in the
most natural form! Maybe we should stick to the simpler polytopes for now :)
Melinda:
When I started typing this message, I wrote your reply first. However I see
since then, Nan has already addressed your question. I might as well back
him up: he is absolutely right. The 120Cell has a large number of vertices
compared to cells, while its dual, the 600Cell has a large number of cells
compared to vertices. Assuming we are talking about cell turning puzzles in
each case, the number of cells determines the "complexity" of each puzzle.
The cut intersections will be more numerous on the 600Cell, less symmetric,
and produce a great deal many more pieces than the 120Cell. However, as I
hinted at above, there is something to be said for the duality. Each of the
120Cell and 600Cell has 7200-fold symmetry. Since a given piece can only
occupy a space where it holds the same geometrical relationship to the cells
of the puzzle, there can only be a maximum of 7200 instances of a single
type of piece on either puzzle (if the piece displays the minimum amount of
symmetry while remaining achiral: chiral pieces may exist in separate orbits
where each orbit consists of 7200 identical pieces and the argument could be
made that there are 14400 instances of that piece type, although half are
mirror images of the other half). Anyway, a 600Cell face turning puzzle
would be quite formidable indeed, and much more difficult than the 120Cell.
Nan:
Your intuition serves you well. I can already tell a cell-turning 600Cell
puzzle would be nothing short of apocalyptic ;)
And if you start looking at deeper cuts, we're no longer talking about
thousands of copies of dozens of types of pieces. We are talking tens of
thousands of copies of hundreds of types of pieces. I would like to state
right now that I will NOT be attempting a cell-turning 600Cell solve without
use of a programmable interface that can locate and execute hundreds of
setups and algorithms for me! (I have no problem with identifying the
algorithms though!)
Thank you all for such an interesting discussion
Keep it up!
-Matt Galla
PS My spellchecker does not recognize the word achiral ----- that's
acceptable
My spellchecker also does not recognize the word vertices ------ REALLY?!?!
sigh...
PPS Following Melinda's example, I also deleted the quoted sections to cut
down on length, which I know for me can be an issue... :)
--00151749f6a2622120049abd9f37
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eel some of the excitement I've had for a long time :)
thanks! And yes, I think you are associating the extra cut to the right phe=
nomenon. Now there may be a more mathematical reason for this, but my impre=
ssion is that these extra cuts are caused by the number of cells meeting at=
a edge and at a vertex in the 4D shape. In the case of the 24Cell, 3 cells=
meet at every edge (and 2 faces meet at every edge in an octahedron). When=
one of these cells is rotated, it also moves one layer of each of the othe=
r 2 cells, while the pieces "in common" of these two layers are p=
airs of stickers that represent the same group of pieces. However, 6 cells =
meet at every vertex (and 4 faces meet at every vertex in an octahedron). W=
hen one of these cells is rotated, it also moves one layer of 4 of the othe=
r cells (see the correlations in the numbers?). However, the remaining cell=
is also affected and it is from this interaction the additional cut is bor=
n, parallel to the vertex (and it must line up with the cut depth of the cu=
ts parallel to the faces of the cell). On the 16Cell, 4 cells meet at an ed=
ge, leaving one that receives a new cut parallel to the edge; and 8 cells m=
eet at a vertex: moving one moves 3 via face cuts, and 3 via edge cuts. The=
remaining 1 receives a new cut parallel to the vertex. So yes, a cell-turn=
ing 16Cell exhibits all 3 cut types. I also agree with your hesitation on t=
he ambiguous names for these monsters. I was merely using 4D Skewb and 24Ce=
ll FTO in lieu of a better name for discussion purposes. Though I am having=
trouble seeing how 4D megaminx can possibly refer to anything but the Magi=
c120Cell.... Perhaps I need to ponder that longer.
7 pieces in one cell: 6 corners (6C), 6 sub-corners (1C), 12 edges (3C), 24=
"offset faces" (2C), 8 "sub-faces" (!!!!!!!2C!!!!!!!),=
and 1 center (1C) [grand total of 672 visible pieces for the whole puzzle,=
I believe]. For this puzzle the sub-corners are not mathematically connect=
ed in any way, so each one acts independently and would naturally have only=
one color. However, what I am calling the "sub-faces" (attemptin=
g to use your terminology) ARE mathematically connected in pairs (in fact I=
believe they would be one physical piece if the 4D puzzle could be constru=
cted in physical form). Because of this, each piece has 2 colors (stickers)=
, making=A0their orientation partially distinguishable: only=A03 of the 6 p=
ossible orientations would be accepted as correct.=A0Adding extra colors to=
make the orientation (and permutation in the case of the sub-corners!) of =
these pieces visible is of course an option, just as it was on the 4D cube =
and 120Cell with the centers. As neither of those puzzles used them, I was =
sort of leaning toward not adding "super" colors, but there is de=
finitely something missing without them. Maybe it could be optional to add =
"super stickers"? :)
on your cut depth. I have to admit, I was too interested in the 24Cell to r=
eally consider any tetrahedral-faced puzzle but I certainly can now. Assumi=
ng a cut shallow enough that no deeper interactions have occurred, I believ=
e you get a few more pieces than you have listed. Applying all 3 types of c=
uts, I THINK I am seeing:
,
2C),
] (1C),=A0=A0 (these are tricky - they are located between=
2 "offset face" stickers and 1 edge sticker)
sub-corner sticker and a center sticker)
ts is impossible if x<y or y<z at shallow depths, and x=3Dy only at v=
ertices so I THINK this is all of the available interactions]
he whole puzzle, I believe] That's almost as bad as the 24Cell ;) Also =
as near as I can tell, increasing depth changes nothing until the center st=
icker is squeezed out of existence (or, more accurately, the 3D surface of =
that cell). After that, things get complicated....
ean by diagonal?) cuts would be a 4D shape mod of a cell turning 16Cell *sh=
udders at thought of 4D shape-modding....*=A0 Depending on the exact nature=
of the geometry (I haven't looked into it yet), a few pieces may be lo=
st or gained going from one puzzle to the other, but proper depths should p=
reserve most piece types between the two puzzles.
e interesting 4D puzzles without getting as big and repetitive as the 120Ce=
ll. Don't you think the 600Cell is headed in the wrong direction? ;) Th=
e symmetry guarantees there will be no more than 14400 of any chiral pieces=
and no more than 7200 of any non-chiral pieces (the same guarantee we have=
for cell-turning 120Cell puzzles), but still, the numbers can get ridiculo=
us really fast on a 600Cell :)
are=A05 different cuts appearing. In addition to the standard face cut, th=
ere are=A0a pair of angled, mirrored cuts due to edge=A0interactions=A0(tha=
t can probably be considered only one type of cut) and 4 types of cuts due =
to vertex interactions (of which 2 are mirror images of each other and prob=
ably best considered as only one type). This gives types of cuts that are n=
ot set parallel to the faces of each cell in addition to the one type that =
is set parallel to each face. Not only would this puzzle have an obscene nu=
mber of each type of piece. There will be dozens of different piece types i=
n the most natural form! Maybe we should stick to the simpler polytopes for=
now :)
I see since then, Nan has already addressed your question. I might as well =
back him up: he is absolutely right. The 120Cell has a large number of vert=
ices compared to cells, while its dual, the 600Cell has a large number of c=
ells compared to vertices. Assuming we are talking about cell turning puzzl=
es in each case, the number of cells determines the "complexity" =
of each puzzle. The cut intersections will be more numerous on the 600Cell,=
less symmetric, and produce a great deal many more pieces than the 120Cell=
. However, as I hinted at above, there is something to be said for the dual=
ity. Each of the 120Cell and 600Cell has 7200-fold symmetry. Since a given =
piece can only occupy a space where it holds=A0the same geometrical relatio=
nship to the cells of the puzzle, there can only be a maximum of 7200 insta=
nces of a single type of piece on either puzzle (if the piece displays the =
minimum amount of symmetry=A0while=A0remaining=A0achiral: chiral pieces may=
exist in separate orbits where each orbit consists of 7200 identical piece=
s and the argument could be made that there are 14400 instances of that pie=
ce type, although half are mirror images of the other half). Anyway, a 600C=
ell face turning puzzle would be quite formidable indeed, and much more dif=
ficult than the 120Cell.
n already tell a cell-turning 600Cell puzzle would be nothing short of apoc=
alyptic ;)
bout thousands of copies of dozens of types of pieces. We are talking tens =
of thousands of copies of hundreds of types of pieces. I would like to stat=
e right now that I will NOT be attempting a cell-turning 600Cell solve with=
out use of a programmable interface that can locate and execute hundreds of=
setups and algorithms for me! (I have no problem with identifying the algo=
rithms though!)
g discussion
39;s acceptable
Y?!?! sigh...
I also deleted the quoted sections to cut down on length, which I know for=
me can be an issue... :)
--00151749f6a2622120049abd9f37--
From: "Andrey" <andreyastrelin@yahoo.com>
Date: Wed, 26 Jan 2011 11:34:16 -0000
Subject: [MC4D] Re: Other 4D puzzles
--- In 4D_Cubing@yahoogroups.com, "Galla, Matthew"
> Thanks for really running with this idea! I hope you are starting to feel
> some of the excitement I've had for a long time :)
I usually think about these small puzzles during my walks with the dog :)
As for 600-cell (and 16-cell), we can consider impossball-like cuttings - w=
ith only 15 stickers per tetrahedron. So we'll have one piece for each corn=
er, edge and face (20C, 5C and 2C in the case of 600-cell). It may be easy =
enough. 16-cell (with its cube-form vertices) may be even solvable :)
It's interesting, but MS5D is waiting. Time to design macros autoreference =
for it :D
Andrey
From: "Andrey" <andreyastrelin@yahoo.com>
Date: Wed, 26 Jan 2011 13:55:04 -0000
Subject: [MC4D] Re: Other 4D puzzles
--- In 4D_Cubing@yahoogroups.com, "Galla, Matthew"
>=20
> As far as the face-turning 16 cell goes, the number of pieces depends on
> your cut depth. I have to admit, I was too interested in the 24Cell to
> really consider any tetrahedral-faced puzzle but I certainly can now.
> Assuming a cut shallow enough that no deeper interactions have occurred, =
I
> believe you get a few more pieces than you have listed. Applying all 3 ty=
pes
> of cuts, I THINK I am seeing:
> 4 corners [rotated by 3 face cuts, 3 edge cuts, and 1 vertex cut] (8C),
> 4 sub-corners [rotated by 3 face cuts and 3 edge cuts] (1C),
> 12 "offset faces" [rotated by 3 face cuts and 2 edge cuts] (2C),
> 12 "offset sub-edges" [rotated by 3 face cuts and 1 edge cut] (1C), (th=
ese
> are tricky - they are located between 2 "offset face" stickers and 1 edge
> sticker)
> 4 "sub-sub-corners" [rotated by 3 face cuts] (1C) (Dunno what else to cal=
l
> these - they are located exactly between a sub-corner sticker and a cente=
r
> sticker)
> 6 edges [rotated by 2 face cuts and 1 edge cut] (4C),
> 6 sub-edges [rotated by 2 face cuts] (1C)
> 4 faces [rotated by 1 facecut] (2C)
> 1 center [rotated by nothing but cell rotation itself] (1C)
>=20
> [it appears that rotation by x face cuts, y edge cuts, and z vertex cuts =
is
> impossible if x
> THINK this is all of the available interactions]
>=20
> That gives a total of 53 stickers per cell! [for 592 pieces total in the
> whole puzzle, I believe] That's almost as bad as the 24Cell ;)=20
Yes, I've checked again - it's really 53. I've missed intersections that ar=
e twice far from the vertex than vertex-cutting plane. So there are 9 stick=
ers assotiated with the vertex, not 2 as I thought before.
> Also as near
> as I can tell, increasing depth changes nothing until the center sticker =
is
> squeezed out of existence (or, more accurately, the 3D surface of that
> cell). After that, things get complicated....
Yes, it will be at depth=3D1/4 of the cell height :)
>=20
> And yes a hypercube with "Skewb-like" (supposedly what you mean by
> diagonal?) cuts would be a 4D shape mod of a cell turning 16Cell *shudder=
s
> at thought of 4D shape-modding....* Depending on the exact nature of the
> geometry (I haven't looked into it yet), a few pieces may be lost or gain=
ed
> going from one puzzle to the other, but proper depths should preserve mos=
t
> piece types between the two puzzles.
Even when pieces are preserved, they may change coloring - from 1C to 8C an=
d back. With all orientation problems :)=20
From: Roice Nelson <roice3@gmail.com>
Date: Wed, 26 Jan 2011 11:20:55 -0600
Subject: Re: [MC4D] Re: Other 4D puzzles
--001636457ce44b1df5049ac30ebd
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in it, 24-cell, 16-cell, 600-cell, and lots more, even the grand antiprism!
If you've never seen the last, I highly recommend checking out the wikipedia
page
strange hybrid between the 120-cell (with two rings of pentagonal antiprisms
playing the role of rings of dodecahedra) and 600-cell (it has 300
tetrahedra). However, even though his engine supports the shapes and
generally positioned slicing, he hasn't actually sliced up any of the
polytopes with non-simplex vertex figures yet. I think there is probably a
good deal of hidden work there, and that it is a "90% done, 90% to go" kind
of situation. Generic implementation of twisting and slicing for the
stranger polytopes surely can't be trivial.
Thanks Nan for the FTO links! I ordered one from the same site Melinda did,
and am already excited for it to arrive :D
Melinda and Matt, I agree the 120-cell is the most natural analogue to the
dodecahedron (agree there is no other 4D polytope the dodecahedron could
better be associated with, nor any other 3D shape that would deserve to be
associated with the 120-cell). The Magic120Cell page even says "4D
Megaminx." I guess my hesitation for the term had more to do with the
dramatic change in properties from 3D to 4D, contrasted with the cube or
simplex situations where properties change in more straightforward and
sequential ways. I hadn't really put my finger on my discomfort, but now
realize part of it was a bias against the idea of an analogue being created
by repeating faces rather than stretching into the next dimension. I think
both your reactions and a little further reflection have removed that small
discomfort though, so I take it back :)
As an aside, there is a short paper titled "The Story of the
120-Cell
which describes what may be the most insane of the connections between the
dodecahedron and the 120-cell. If you interpret the 4D vector locations of
the cell centers of the 120-cell as quaternions, the 3D rotations those
quaternions describe are none other than the symmetries of the dodecahedron
(and icosahedron). Likewise, the 24-cell encodes the symmetries of the
tetrahedron. This all really shocked me when I first read it, and was
almost enough to make me flirt with believing in intelligent design :)
Cheers,
Roice
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/span>Don's latest engine does have =
all the polytopes in it, 24-cell, 16-cell, 600-cell, and lots more, even th=
e grand antiprism!=A0 If you've never seen the last, I highly recommend=
checking out the =
wikipedia page on it.=A0 It's sort of a strange hybrid between the =
120-cell (with two rings of pentagonal antiprisms playing the role of rings=
of dodecahedra) and 600-cell (it has 300 tetrahedra).=A0 However, even tho=
ugh his engine supports the shapes and generally positioned slicing, he has=
n't actually sliced up any of the polytopes with non-simplex vertex fig=
ures yet.=A0 I think there is probably a good deal of hidden work there, an=
d that it is a "90% done, 90% to go" kind of situation.=A0=A0Gene=
ric=A0implementation of=A0twisting and slicing for the stranger polytopes s=
urely can't be trivial.
Thanks Nan for the FTO links!=A0 I ordered one from the same site Melin=
da did, and am already excited for it to arrive :D
Melinda and Matt,=
I agree the 120-cell is the most natural analogue to the dodecahedron (agr=
ee there is no other 4D polytope the dodecahedron could better be associate=
d with, nor any other 3D shape that would deserve to be associated with the=
120-cell).=A0 The Magic120Cell page even says "4D Megaminx."=A0 =
I guess my hesitation for the term had more to do with the dramatic change =
in properties from 3D to 4D, contrasted with the cube or simplex situations=
where properties change in more straightforward and sequential=A0ways.=A0 =
I hadn't really put my finger on my discomfort, but now realize part of=
it was a bias against the idea of an analogue being created by repeating f=
aces rather than stretching into the next dimension.=A0 I think both your r=
eactions and a little further reflection have removed that small discomfort=
though, so I take it back :)
As an aside,=A0there is a=A0short paper titled "www.ams.org/notices/200101/fea-stillwell.pdf">The Story of the 120-Cell=
" which describes what may be the most insane of the connections betwe=
en the dodecahedron and the 120-cell.=A0 If you interpret the 4D vector loc=
ations of the cell centers of the 120-cell as quaternions, the 3D rotations=
those quaternions describe are none other than the symmetries of the dodec=
ahedron (and icosahedron).=A0 Likewise, the 24-cell encodes the symmetries =
of the tetrahedron.=A0 This all really shocked me when I first read it, and=
was almost enough to make me flirt with believing in intelligent design :)=
Roice
--001636457ce44b1df5049ac30ebd--
From: Melinda Green <melinda@superliminal.com>
Date: Wed, 26 Jan 2011 12:08:24 -0800
Subject: Re: [MC4D] Re: Other 4D puzzles
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Don't give in to mysticism, Roice. :-) The best antidote I've found is
this wonderful paper
once-and-for-all clearly explains the connection between mathematics and
the rest of science including all those strange and marvelous
connections we keep finding. It is the single most profound thing I have
ever read. It is short and easy and fun to read. Read it carefully,
because like a favorite movie, you only get to experience it for the
first time once.
-Melinda
On 1/26/2011 9:20 AM, Roice Nelson wrote:
>
> [...]
>
> As an aside, there is a short paper titled "The Story of the 120-Cell
>
> what may be the most insane of the connections between the
> dodecahedron and the 120-cell. If you interpret the 4D vector
> locations of the cell centers of the 120-cell as quaternions, the 3D
> rotations those quaternions describe are none other than the
> symmetries of the dodecahedron (and icosahedron). Likewise, the
> 24-cell encodes the symmetries of the tetrahedron. This all really
> shocked me when I first read it, and was almost enough to make me
> flirt with believing in intelligent design :)
--------------090000040101080302090802
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit
http-equiv="Content-Type">
Don't give in to mysticism, Roice. :-) The best antidote I've found
is this
wonderful paper that once-and-for-all clearly explains the
connection between mathematics and the rest of science including all
those strange and marvelous connections we keep finding. It is the
single most profound thing I have ever read. It is short and easy
and fun to read. Read it carefully, because like a favorite movie,
you only get to experience it for the first time once.
-Melinda
On 1/26/2011 9:20 AM, Roice Nelson wrote:
type="cite">
[...]
As an aside, there is a short paper titled "
href="http://www.ams.org/notices/200101/fea-stillwell.pdf">The
Story of the 120-Cell" which describes what may be the
most insane of the connections between the dodecahedron and the
120-cell. If you interpret the 4D vector locations of the cell
centers of the 120-cell as quaternions, the 3D rotations those
quaternions describe are none other than the symmetries of the
dodecahedron (and icosahedron). Likewise, the 24-cell encodes
the symmetries of the tetrahedron. This all really shocked me
when I first read it, and was almost enough to make me flirt
with believing in intelligent design :)
--------------090000040101080302090802--
From: Melinda Green <melinda@superliminal.com>
Date: Wed, 26 Jan 2011 17:18:16 -0800
Subject: Re: [MC4D] Re: Other 4D puzzles
Andrey,
That is an ambitious goal and a very noble one. I agree completely that
the log file is the right place to start with any of this. It is the
"document" or "model" in object-oriented design and is therefore at the
core of this whole endeavor. I hope that you make sure to support all
the features of the MC4D implementation so that we'll be prepared to
switch to your system someday.
One thing that I regret not bringing from v1 to v2 of the MC4D log
format is the state block. That is useful for passing around starting
configurations that do not contain the instructions for how it was
scrambled. I encourage you to make accordance for that and other sorts
of optional data blocks.
Thanks,
-Melinda
On 1/23/2011 11:49 AM, Andrey wrote:
> Matt,
> My plan after 5D simplex is some 4D puzzle. It may be 24cell, or hypercube with some different cutting (1C-centered and 4C-centered? don't know). For the twists I'm going to use free rotation of layer: you select it (face by left-click, 2C/3C/4C if there are such centers - by right click) and then use free rotation in 3D around the fixed center. I also thought about additional stickers - but it's equivalent to work with some truncated puzzle: additional faces will play the role of stickers for non-face-centered layers.
> But before there will be time to write description of log files for the wide class of 4D puzzles. I think that it must include:
> - symmetry of the body (set of possible rotation axes)
> - number of layers for each class of axes (e.g. hypercube, 3 layers in 1C direction, 2 layers in 4C direction)
> - sequence of twists in terms "direction of rotation center, direction of rotation axis, angle, layer mask"
>
> All the other things (actual shape of main body, coloring, depths of cuts) are not so important - they are about the puzzle description and if two programs will recognize, say, the same name in the similar way, they will give same interpretation of the rest of log file.
>
> Andrey
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>