Thread: "Fractal cubes"

Have a look at the following image. It brings together two of my
favorite subjects: Fractals and Rubik's cubes.


http://proofmathisbeautiful.tumblr.com/photo/1280/431394685/1/tumblr_kyv2avNAWg1qa3dmv

I have no idea what's behind it. Has someone actually built a fractal
twisty puzzle? If not, would that be possible? I expect that a working
fractal cube would feel very much like a 3D version of Roice's Magic
Tile in which each area is repeated across space but at multiple scales
in addition to translations and rotations.

Fascinated,
-Melinda

This is just a typical fractal generation with a highly
regular algorithm. (I am trying to distinguish it from the
more interesting fractals (think of coastlines) that exhibit
randomness.) Instead of going smaller on each iteration,
the pattern becomes larger in this case. The basic starting
pattern is a pile of 20 cubes, corresponding to a 3x3x3
stack with the central 'cross' (7 cubes) removed. That
stack, with the holes in it, can be treated as a cube
itself. So 20 such cubical piles can be piled together in
the analogous fashion to create the next generation - a pile
of 400 little cubes. Etc.

The coloring looks random to me. (I can imagine interesting
looking non-random colorings, some of which could improve
one's ability to see the picture correctly.)

As an abstract thing, the 20-cube pile could be 'worked'
like a 3D puzzle. (I have a recollection that there is a
commercial physical version of such a puzzle.) The only
catch is that, in the absence of face-center pieces, you
have to use some other method to assign the face colors.
However, this is already a familiar problem with the even
order puzzles. If you use the analogous motions to 'work'
a 400-cube pile, you see that little cubes can never move
from one 20-cube pile to another; so that does not lead to
an especially interesting puzzle. OTOH, the 400-cube
pile could be regarded as a variation on the order-9
3-puzzle; and this one is interesting, as we can see
cubies that are not in external slices. I.e., we can begin
to concern ourselves with the permutation and orientation
of interior cubies that are normally invisible to us.

Melinda, how did you encounter this? Surely there must
be some context that would provide a little more info about
its significance (or lack thereof).

Regards,
David V.


----- Original Message -----
From: "Melinda Green"
To: "MagicCube4D" <4D_Cubing@yahoogroups.com>
Sent: Sunday, March 07, 2010 9:59 PM
Subject: [MC4D] Fractal cubes


Have a look at the following image. It brings together two of my
favorite subjects: Fractals and Rubik's cubes.


http://proofmathisbeautiful.tumblr.com/photo/1280/431394685/1/tumblr_kyv2avNAWg1qa3dmv

I have no idea what's behind it. Has someone actually built a
fractal
twisty puzzle? If not, would that be possible? I expect that a
working
fractal cube would feel very much like a 3D version of Roice's
Magic
Tile in which each area is repeated across space but at multiple
scales
in addition to translations and rotations.

Fascinated,
-Melinda

I managed to find some context:
http://dosenjp.tumblr.com/post/430499178/via-dothereject

There is there a comment indicating that this thing has a name -
a Menger sponge:
http://en.wikipedia.org/wiki/Menger_sponge
If you found my explanation of its 'construction' too terse,
there is a much
more elaborate version on the wiki.

So the image was basically a coloring of a Menger sponge in the
manner
of a Rubik's Cube.

Regards,
David V.

David,

I'm aware of Menger cubes, Serpenski gaskets, Cantor dust, etc. Their
constructions are quite simple. Normally quite boringly simple, but I
find this coloration of a Menger cube really quite evocative. I don't
have any more context for the image. I just get Google alerts on the
term "Buddhabrot" which lately includes a lot of postings such as this
one by a person who used that term as their account name. Quite
flattering really and I often like what they come up with. Yes, the
image appears to be a random Rubik colored Menger cube and not any sort
of actual puzzle. It just makes me wonder how it could best be made real.

My current thought is to treat every 20-cube figure identically,
regardless of scale. So a twist on a 20 "atom" cube would cause a twist
of every other 20 atom cube as well as every 400 atom cube and so on up
the chain to some maximum scale cube. Scrambling such a beast would mix
the colors so completely that at high levels it will just appear as a
single grainy mix of all the colors. You'd need to zoom in to the atomic
level in order to work on it. In some ways, the fundamental puzzle would
not be terribly interesting because I think that it would just be normal
void cube (http://en.wikipedia.org/wiki/Void_Cube) because as you point
out, the cubies would not mix outside their respective cubes at any
level. Still, it'd be fascinating to watch it being solved.

Does anyone else have ideas for better ways to make this thing real? The
only constraint that I urge is that a twist on any scale should have
identical affects on all scales, but just how that might work is an open
question.

-Melinda


David Vanderschel wrote:
> This is just a typical fractal generation with a highly
> regular algorithm. (I am trying to distinguish it from the
> more interesting fractals (think of coastlines) that exhibit
> randomness.) Instead of going smaller on each iteration,
> the pattern becomes larger in this case. The basic starting
> pattern is a pile of 20 cubes, corresponding to a 3x3x3
> stack with the central 'cross' (7 cubes) removed. That
> stack, with the holes in it, can be treated as a cube
> itself. So 20 such cubical piles can be piled together in
> the analogous fashion to create the next generation - a pile
> of 400 little cubes. Etc.
>
> The coloring looks random to me. (I can imagine interesting
> looking non-random colorings, some of which could improve
> one's ability to see the picture correctly.)
>
> As an abstract thing, the 20-cube pile could be 'worked'
> like a 3D puzzle. (I have a recollection that there is a
> commercial physical version of such a puzzle.) The only
> catch is that, in the absence of face-center pieces, you
> have to use some other method to assign the face colors.
> However, this is already a familiar problem with the even
> order puzzles. If you use the analogous motions to 'work'
> a 400-cube pile, you see that little cubes can never move
> from one 20-cube pile to another; so that does not lead to
> an especially interesting puzzle. OTOH, the 400-cube
> pile could be regarded as a variation on the order-9
> 3-puzzle; and this one is interesting, as we can see
> cubies that are not in external slices. I.e., we can begin
> to concern ourselves with the permutation and orientation
> of interior cubies that are normally invisible to us.
>
> Melinda, how did you encounter this? Surely there must
> be some context that would provide a little more info about
> its significance (or lack thereof).
>
> I managed to find some context:
> http://dosenjp.tumblr.com/post/430499178/via-dothereject
>
> There is there a comment indicating that this thing has a name -
> a Menger sponge:
> http://en.wikipedia.org/wiki/Menger_sponge
> If you found my explanation of its 'construction' too terse,
> there is a much
> more elaborate version on the wiki.
>
> So the image was basically a coloring of a Menger sponge in the
> manner
> of a Rubik's Cube.
>

--001485f8139e7963840481581253
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Content-Transfer-Encoding: quoted-printable

Well, since the void cube is a level 1 Menger sponge, it would always be
possible to go simply to just level 2 for now and have a finite puzzle in
that sense. I don't know if it would be possible to construct a physical
one without having it rounded in shape like the V-Cubes, but on a computer
it would be no problem. How would it work? Let's say you take the front
upper middle edge cube (almost equivalent to a void cube). You would be
able to do L, M, and R twists on this as much as you want, but if you try,
say, a U twist, you will end up twisting the entire upper layer. This woul=
d
move the top 8 cubies of the front upper middle cube and move it to the top
8 cubies of one of the side upper middle cubes, and similarly for other
twists like it. This would give the cube a crazy level of
interconnectedness between all the mini void cubes and would make for one
crazy puzzle to solve. This method could also be extended to any level of
depth desired, except beyond level 2 the number of cubies just becomes waaa=
y
too big for any sane person to deal with :D

As for coloring of the inside stickers (the void cube is black inside but
the Menger sponge has to have inside colors at least above level 1 but it's
no more difficult to have for all cubies) just have the same color as the
whole cube. So you would still have 6 colors, but if you look at one of th=
e
edge pieces of a mini void cube, instead of being 2 colored, it would be 4
colored. But because of the way the coloring works, the 4 coloredness of
the cubie is equivalent to the normal 2 colored edge pieces.

Chris

2010/3/9 Melinda Green

>
>
> David,
>
> I'm aware of Menger cubes, Serpenski gaskets, Cantor dust, etc. Their
> constructions are quite simple. Normally quite boringly simple, but I
> find this coloration of a Menger cube really quite evocative. I don't
> have any more context for the image. I just get Google alerts on the
> term "Buddhabrot" which lately includes a lot of postings such as this
> one by a person who used that term as their account name. Quite
> flattering really and I often like what they come up with. Yes, the
> image appears to be a random Rubik colored Menger cube and not any sort
> of actual puzzle. It just makes me wonder how it could best be made real.
>
> My current thought is to treat every 20-cube figure identically,
> regardless of scale. So a twist on a 20 "atom" cube would cause a twist
> of every other 20 atom cube as well as every 400 atom cube and so on up
> the chain to some maximum scale cube. Scrambling such a beast would mix
> the colors so completely that at high levels it will just appear as a
> single grainy mix of all the colors. You'd need to zoom in to the atomic
> level in order to work on it. In some ways, the fundamental puzzle would
> not be terribly interesting because I think that it would just be normal
> void cube (http://en.wikipedia.org/wiki/Void_Cube) because as you point
> out, the cubies would not mix outside their respective cubes at any
> level. Still, it'd be fascinating to watch it being solved.
>
> Does anyone else have ideas for better ways to make this thing real? The
> only constraint that I urge is that a twist on any scale should have
> identical affects on all scales, but just how that might work is an open
> question.
>
> -Melinda
>
>
> David Vanderschel wrote:
> > This is just a typical fractal generation with a highly
> > regular algorithm. (I am trying to distinguish it from the
> > more interesting fractals (think of coastlines) that exhibit
> > randomness.) Instead of going smaller on each iteration,
> > the pattern becomes larger in this case. The basic starting
> > pattern is a pile of 20 cubes, corresponding to a 3x3x3
> > stack with the central 'cross' (7 cubes) removed. That
> > stack, with the holes in it, can be treated as a cube
> > itself. So 20 such cubical piles can be piled together in
> > the analogous fashion to create the next generation - a pile
> > of 400 little cubes. Etc.
> >
> > The coloring looks random to me. (I can imagine interesting
> > looking non-random colorings, some of which could improve
> > one's ability to see the picture correctly.)
> >
> > As an abstract thing, the 20-cube pile could be 'worked'
> > like a 3D puzzle. (I have a recollection that there is a
> > commercial physical version of such a puzzle.) The only
> > catch is that, in the absence of face-center pieces, you
> > have to use some other method to assign the face colors.
> > However, this is already a familiar problem with the even
> > order puzzles. If you use the analogous motions to 'work'
> > a 400-cube pile, you see that little cubes can never move
> > from one 20-cube pile to another; so that does not lead to
> > an especially interesting puzzle. OTOH, the 400-cube
> > pile could be regarded as a variation on the order-9
> > 3-puzzle; and this one is interesting, as we can see
> > cubies that are not in external slices. I.e., we can begin
> > to concern ourselves with the permutation and orientation
> > of interior cubies that are normally invisible to us.
> >
> > Melinda, how did you encounter this? Surely there must
> > be some context that would provide a little more info about
> > its significance (or lack thereof).
> >
> > I managed to find some context:
> > http://dosenjp.tumblr.com/post/430499178/via-dothereject
> >
> > There is there a comment indicating that this thing has a name -
> > a Menger sponge:
> > http://en.wikipedia.org/wiki/Menger_sponge
> > If you found my explanation of its 'construction' too terse,
> > there is a much
> > more elaborate version on the wiki.
> >
> > So the image was basically a coloring of a Menger sponge in the
> > manner
> > of a Rubik's Cube.
> >
>=20=20
>

--001485f8139e7963840481581253
Content-Type: text/html; charset=UTF-8
Content-Transfer-Encoding: quoted-printable

Well, since the void cube is a level 1 Menger sponge, it would always be po=
ssible to go simply to just level 2 for now and have a finite puzzle in tha=
t sense.=C2=A0 I don't know if it would be possible to construct a phys=
ical one without having it rounded in shape like the V-Cubes, but on a comp=
uter it would be no problem.=C2=A0 How would it work?=C2=A0 Let's say y=
ou take the front upper middle edge cube (almost equivalent to a void cube)=
.=C2=A0 You would be able to do L, M, and R twists on this as much as you w=
ant, but if you try, say, a U twist, you will end up twisting the entire up=
per layer.=C2=A0 This would move the top 8 cubies of the front upper middle=
cube and move it to the top 8 cubies of one of the side upper middle cubes=
, and similarly for other twists like it.=C2=A0 This would give the cube a =
crazy level of interconnectedness between all the mini void cubes and would=
make for one crazy puzzle to solve.=C2=A0 This method could also be extend=
ed to any level of depth desired, except beyond level 2 the number of cubie=
s just becomes waaay too big for any sane person to deal with :D


As for coloring of the inside stickers (the void cube is black inside b=
ut the Menger sponge has to have inside colors at least above level 1 but i=
t's no more difficult to have for all cubies) just have the same color =
as the whole cube.=C2=A0 So you would still have 6 colors, but if you look =
at one of the edge pieces of a mini void cube, instead of being 2 colored, =
it would be 4 colored.=C2=A0 But because of the way the coloring works, the=
4 coloredness of the cubie is equivalent to the normal 2 colored edge piec=
es.


Chris

We may not be talking about the same basic geometry, but if we are then
I think that I see the problem. When I say that a twist on a level 1
cube (I.E. a Void cube) will affect all other cubes, even of larger
levels, meant for twists on larger level cubes to be in proportion to
those cubes. So in your 2-level version, twisting a 3x3x1 slice will not
only make the same twist on all 19 other level 1 cubes, but will also
twist the corresponding 9x9x3 face of the level 2 cube. Rather than
mixing up the puzzle beyond all hope (for say a 4 level cube), this
design mixes them not at all and seems to leave us with a single void
cube to solve.

Regarding realizations of possible puzzles, I only mean computer
implementations.

Regarding the coloring of inside stickers, I didn't completely follow
what you proposed, but it doesn't seem like an important issue because I
don't think that the inside stickers can contribute to the puzzle. I
think they should automatically be correct when the rest of the puzzle
is complete. I'd therefore probably color them with the same color as
the outside stickers that face in the same direction. That way when the
whole puzzle is solved it will appear to be more "complete", but maybe
that was what you were saying.

So how to define a Rubik's Menger Cube that's harder than a Void cube
yet not impossible? Here's another idea: What if a twist on any level 1
cube twists all other cubes of all levels that are currently playing the
same role? IOW, if you twists a level-1 cube that is part of the FUR
cubie, then all other cubes that are also FUR elements of other cubes
will twist, and *only* those cubes. That might make all cubes into
individually solvable void cubes yet hopelessly interconnected with each
other. I don't know but I sense that there may be a natural, elegant,
and hard-but-not-impossible-puzzle definition lurking here somewhere.

-Melinda

Chris Locke wrote:
>
>
> Well, since the void cube is a level 1 Menger sponge, it would always
> be possible to go simply to just level 2 for now and have a finite
> puzzle in that sense. I don't know if it would be possible to
> construct a physical one without having it rounded in shape like the
> V-Cubes, but on a computer it would be no problem. How would it
> work? Let's say you take the front upper middle edge cube (almost
> equivalent to a void cube). You would be able to do L, M, and R
> twists on this as much as you want, but if you try, say, a U twist,
> you will end up twisting the entire upper layer. This would move the
> top 8 cubies of the front upper middle cube and move it to the top 8
> cubies of one of the side upper middle cubes, and similarly for other
> twists like it. This would give the cube a crazy level of
> interconnectedness between all the mini void cubes and would make for
> one crazy puzzle to solve. This method could also be extended to any
> level of depth desired, except beyond level 2 the number of cubies
> just becomes waaay too big for any sane person to deal with :D
>
> As for coloring of the inside stickers (the void cube is black inside
> but the Menger sponge has to have inside colors at least above level 1
> but it's no more difficult to have for all cubies) just have the same
> color as the whole cube. So you would still have 6 colors, but if you
> look at one of the edge pieces of a mini void cube, instead of being 2
> colored, it would be 4 colored. But because of the way the coloring
> works, the 4 coloredness of the cubie is equivalent to the normal 2
> colored edge pieces.
>
> Chris
>
> 2010/3/9 Melinda Green > >
>
>
>
> David,
>
> I'm aware of Menger cubes, Serpenski gaskets, Cantor dust, etc. Their
> constructions are quite simple. Normally quite boringly simple, but I
> find this coloration of a Menger cube really quite evocative. I don't
> have any more context for the image. I just get Google alerts on the
> term "Buddhabrot" which lately includes a lot of postings such as
> this
> one by a person who used that term as their account name. Quite
> flattering really and I often like what they come up with. Yes, the
> image appears to be a random Rubik colored Menger cube and not any
> sort
> of actual puzzle. It just makes me wonder how it could best be
> made real.
>
> My current thought is to treat every 20-cube figure identically,
> regardless of scale. So a twist on a 20 "atom" cube would cause a
> twist
> of every other 20 atom cube as well as every 400 atom cube and so
> on up
> the chain to some maximum scale cube. Scrambling such a beast
> would mix
> the colors so completely that at high levels it will just appear as a
> single grainy mix of all the colors. You'd need to zoom in to the
> atomic
> level in order to work on it. In some ways, the fundamental puzzle
> would
> not be terribly interesting because I think that it would just be
> normal
> void cube (http://en.wikipedia.org/wiki/Void_Cube) because as you
> point
> out, the cubies would not mix outside their respective cubes at any
> level. Still, it'd be fascinating to watch it being solved.
>
> Does anyone else have ideas for better ways to make this thing
> real? The
> only constraint that I urge is that a twist on any scale should have
> identical affects on all scales, but just how that might work is
> an open
> question.
>
> -Melinda
>
>
>
> David Vanderschel wrote:
> > This is just a typical fractal generation with a highly
> > regular algorithm. (I am trying to distinguish it from the
> > more interesting fractals (think of coastlines) that exhibit
> > randomness.) Instead of going smaller on each iteration,
> > the pattern becomes larger in this case. The basic starting
> > pattern is a pile of 20 cubes, corresponding to a 3x3x3
> > stack with the central 'cross' (7 cubes) removed. That
> > stack, with the holes in it, can be treated as a cube
> > itself. So 20 such cubical piles can be piled together in
> > the analogous fashion to create the next generation - a pile
> > of 400 little cubes. Etc.
> >
> > The coloring looks random to me. (I can imagine interesting
> > looking non-random colorings, some of which could improve
> > one's ability to see the picture correctly.)
> >
> > As an abstract thing, the 20-cube pile could be 'worked'
> > like a 3D puzzle. (I have a recollection that there is a
> > commercial physical version of such a puzzle.) The only
> > catch is that, in the absence of face-center pieces, you
> > have to use some other method to assign the face colors.
> > However, this is already a familiar problem with the even
> > order puzzles. If you use the analogous motions to 'work'
> > a 400-cube pile, you see that little cubes can never move
> > from one 20-cube pile to another; so that does not lead to
> > an especially interesting puzzle. OTOH, the 400-cube
> > pile could be regarded as a variation on the order-9
> > 3-puzzle; and this one is interesting, as we can see
> > cubies that are not in external slices. I.e., we can begin
> > to concern ourselves with the permutation and orientation
> > of interior cubies that are normally invisible to us.
> >
> > Melinda, how did you encounter this? Surely there must
> > be some context that would provide a little more info about
> > its significance (or lack thereof).
> >
> > I managed to find some context:
> > http://dosenjp.tumblr.com/post/430499178/via-dothereject
> >
> > There is there a comment indicating that this thing has a name -
> > a Menger sponge:
> > http://en.wikipedia.org/wiki/Menger_sponge
> > If you found my explanation of its 'construction' too terse,
> > there is a much
> > more elaborate version on the wiki.
> >
> > So the image was basically a coloring of a Menger sponge in the
> > manner
> > of a Rubik's Cube.
> >
>
>
>
>
>

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I've attached an image for clarity.

In the case I was talking of, a single twist won't affect other parts of th=
e
puzzle, just the part that is twisted. So it's like a normal puzzle, just
with a crazy shape. In the image I attached, the green outlines examples o=
f
simple void cube twists that are permitted, and the teal outlines the large=
r
twists that are permitted. Basically, the twisting works just like you
expect it to if a physical level 2 Menger rubik cube existed. The mini
level 1 cubes can be twisted just like a void cube along those axes, but in
the other directions you have to move a larger 9x9x1 slab instead of just a
3x3x1 slice. This means that for the void cubes in the corners of the leve=
l
2 Menger cube, there is no way to just twist the cubies in that individual
mini void cube; all twists are twists of 9x9x1 slabs. This process can be
generalized up to any level as well. So for level 3 Menger cubes, 3x3x1,
9x9x1, and 27x27x1 twists are possible.

As for sticker coloring, if you look at the level 2 Menger cube, you will
notice that the smaller 3x3x3 void cubes act as single cubies at that
level. So if you would color the larger scale level 2 Menger cube simply
like a void cube, the entire inside faces would be uncolored. In the
picture I attached, I blacked out the top face of the front-bottom-middle
void cube to show what that would look like. The only choice would be to
color that blackened face blue, corresponding to the color directly above
it.

You're definition of twisting where one twist propagates seems really
interesting, but hard to visualize :D. Sounds like it could make for a
pretty interesting puzzle. I'd imagine it might be similar to how in
Roice's new MagicTile program the infinite tilings are repeated so a twist
of one face twists all equivalent faces. Only in this case it's not tiling=
,
but 'fractaling' :D. I'll have to think about that a bit more later! Seem=
s
like it would be one heck of a difficult puzzle anyway!

Chris

2010/3/10 Melinda Green

>
>
> We may not be talking about the same basic geometry, but if we are then
> I think that I see the problem. When I say that a twist on a level 1
> cube (I.E. a Void cube) will affect all other cubes, even of larger
> levels, meant for twists on larger level cubes to be in proportion to
> those cubes. So in your 2-level version, twisting a 3x3x1 slice will not
> only make the same twist on all 19 other level 1 cubes, but will also
> twist the corresponding 9x9x3 face of the level 2 cube. Rather than
> mixing up the puzzle beyond all hope (for say a 4 level cube), this
> design mixes them not at all and seems to leave us with a single void
> cube to solve.
>
> Regarding realizations of possible puzzles, I only mean computer
> implementations.
>
> Regarding the coloring of inside stickers, I didn't completely follow
> what you proposed, but it doesn't seem like an important issue because I
> don't think that the inside stickers can contribute to the puzzle. I
> think they should automatically be correct when the rest of the puzzle
> is complete. I'd therefore probably color them with the same color as
> the outside stickers that face in the same direction. That way when the
> whole puzzle is solved it will appear to be more "complete", but maybe
> that was what you were saying.
>
> So how to define a Rubik's Menger Cube that's harder than a Void cube
> yet not impossible? Here's another idea: What if a twist on any level 1
> cube twists all other cubes of all levels that are currently playing the
> same role? IOW, if you twists a level-1 cube that is part of the FUR
> cubie, then all other cubes that are also FUR elements of other cubes
> will twist, and *only* those cubes. That might make all cubes into
> individually solvable void cubes yet hopelessly interconnected with each
> other. I don't know but I sense that there may be a natural, elegant,
> and hard-but-not-impossible-puzzle definition lurking here somewhere.
>
> -Melinda
>
>
> Chris Locke wrote:
> >
> >
> > Well, since the void cube is a level 1 Menger sponge, it would always
> > be possible to go simply to just level 2 for now and have a finite
> > puzzle in that sense. I don't know if it would be possible to
> > construct a physical one without having it rounded in shape like the
> > V-Cubes, but on a computer it would be no problem. How would it
> > work? Let's say you take the front upper middle edge cube (almost
> > equivalent to a void cube). You would be able to do L, M, and R
> > twists on this as much as you want, but if you try, say, a U twist,
> > you will end up twisting the entire upper layer. This would move the
> > top 8 cubies of the front upper middle cube and move it to the top 8
> > cubies of one of the side upper middle cubes, and similarly for other
> > twists like it. This would give the cube a crazy level of
> > interconnectedness between all the mini void cubes and would make for
> > one crazy puzzle to solve. This method could also be extended to any
> > level of depth desired, except beyond level 2 the number of cubies
> > just becomes waaay too big for any sane person to deal with :D
> >
> > As for coloring of the inside stickers (the void cube is black inside
> > but the Menger sponge has to have inside colors at least above level 1
> > but it's no more difficult to have for all cubies) just have the same
> > color as the whole cube. So you would still have 6 colors, but if you
> > look at one of the edge pieces of a mini void cube, instead of being 2
> > colored, it would be 4 colored. But because of the way the coloring
> > works, the 4 coloredness of the cubie is equivalent to the normal 2
> > colored edge pieces.
> >
> > Chris
> >
> > 2010/3/9 Melinda Green .com>
> > >>
>
> >
> >
> >
> > David,
> >
> > I'm aware of Menger cubes, Serpenski gaskets, Cantor dust, etc. Their
> > constructions are quite simple. Normally quite boringly simple, but I
> > find this coloration of a Menger cube really quite evocative. I don't
> > have any more context for the image. I just get Google alerts on the
> > term "Buddhabrot" which lately includes a lot of postings such as
> > this
> > one by a person who used that term as their account name. Quite
> > flattering really and I often like what they come up with. Yes, the
> > image appears to be a random Rubik colored Menger cube and not any
> > sort
> > of actual puzzle. It just makes me wonder how it could best be
> > made real.
> >
> > My current thought is to treat every 20-cube figure identically,
> > regardless of scale. So a twist on a 20 "atom" cube would cause a
> > twist
> > of every other 20 atom cube as well as every 400 atom cube and so
> > on up
> > the chain to some maximum scale cube. Scrambling such a beast
> > would mix
> > the colors so completely that at high levels it will just appear as a
> > single grainy mix of all the colors. You'd need to zoom in to the
> > atomic
> > level in order to work on it. In some ways, the fundamental puzzle
> > would
> > not be terribly interesting because I think that it would just be
> > normal
> > void cube (http://en.wikipedia.org/wiki/Void_Cube) because as you
> > point
> > out, the cubies would not mix outside their respective cubes at any
> > level. Still, it'd be fascinating to watch it being solved.
> >
> > Does anyone else have ideas for better ways to make this thing
> > real? The
> > only constraint that I urge is that a twist on any scale should have
> > identical affects on all scales, but just how that might work is
> > an open
> > question.
> >
> > -Melinda
> >
> >
> >
> > David Vanderschel wrote:
> > > This is just a typical fractal generation with a highly
> > > regular algorithm. (I am trying to distinguish it from the
> > > more interesting fractals (think of coastlines) that exhibit
> > > randomness.) Instead of going smaller on each iteration,
> > > the pattern becomes larger in this case. The basic starting
> > > pattern is a pile of 20 cubes, corresponding to a 3x3x3
> > > stack with the central 'cross' (7 cubes) removed. That
> > > stack, with the holes in it, can be treated as a cube
> > > itself. So 20 such cubical piles can be piled together in
> > > the analogous fashion to create the next generation - a pile
> > > of 400 little cubes. Etc.
> > >
> > > The coloring looks random to me. (I can imagine interesting
> > > looking non-random colorings, some of which could improve
> > > one's ability to see the picture correctly.)
> > >
> > > As an abstract thing, the 20-cube pile could be 'worked'
> > > like a 3D puzzle. (I have a recollection that there is a
> > > commercial physical version of such a puzzle.) The only
> > > catch is that, in the absence of face-center pieces, you
> > > have to use some other method to assign the face colors.
> > > However, this is already a familiar problem with the even
> > > order puzzles. If you use the analogous motions to 'work'
> > > a 400-cube pile, you see that little cubes can never move
> > > from one 20-cube pile to another; so that does not lead to
> > > an especially interesting puzzle. OTOH, the 400-cube
> > > pile could be regarded as a variation on the order-9
> > > 3-puzzle; and this one is interesting, as we can see
> > > cubies that are not in external slices. I.e., we can begin
> > > to concern ourselves with the permutation and orientation
> > > of interior cubies that are normally invisible to us.
> > >
> > > Melinda, how did you encounter this? Surely there must
> > > be some context that would provide a little more info about
> > > its significance (or lack thereof).
> > >
> > > I managed to find some context:
> > > http://dosenjp.tumblr.com/post/430499178/via-dothereject
> > >
> > > There is there a comment indicating that this thing has a name -
> > > a Menger sponge:
> > > http://en.wikipedia.org/wiki/Menger_sponge
> > > If you found my explanation of its 'construction' too terse,
> > > there is a much
> > > more elaborate version on the wiki.
> > >
> > > So the image was basically a coloring of a Menger sponge in the
> > > manner
> > > of a Rubik's Cube.
> > >
> >
> >
> >
> >
> >
>=20=20
>

--0016361e7ba2b4679204816dabd2
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I've attached an image for clarity.

In the case I was talking of=
, a single twist won't affect other parts of the puzzle, just the part =
that is twisted.=C2=A0 So it's like a normal puzzle, just with a crazy =
shape.=C2=A0 In the image I attached, the green outlines examples of simple=
void cube twists that are permitted, and the teal outlines the larger twis=
ts that are permitted.=C2=A0 Basically, the twisting works just like you ex=
pect it to if a physical level 2 Menger rubik cube existed.=C2=A0 The mini =
level 1 cubes can be twisted just like a void cube along those axes, but in=
the other directions you have to move a larger 9x9x1 slab instead of just =
a 3x3x1 slice.=C2=A0 This means that for the void cubes in the corners of t=
he level 2 Menger cube, there is no way to just twist the cubies in that in=
dividual mini void cube; all twists are twists of 9x9x1 slabs.=C2=A0 This p=
rocess can be generalized up to any level as well.=C2=A0 So for level 3 Men=
ger cubes, 3x3x1, 9x9x1, and 27x27x1 twists are possible.


As for sticker coloring, if you look at the level 2 Menger cube, you wi=
ll notice that the smaller 3x3x3 void cubes act as single cubies at that le=
vel.=C2=A0 So if you would color the larger scale level 2 Menger cube simpl=
y like a void cube, the entire inside faces would be uncolored.=C2=A0 In th=
e picture I attached, I blacked out the top face of the front-bottom-middle=
void cube to show what that would look like.=C2=A0 The only choice would b=
e to color that blackened face blue, corresponding to the color directly ab=
ove it.


You're definition of twisting where one twist propagates seems real=
ly interesting, but hard to visualize :D.=C2=A0 Sounds like it could make f=
or a pretty interesting puzzle.=C2=A0 I'd imagine it might be similar t=
o how in Roice's new MagicTile program the infinite tilings are repeate=
d so a twist of one face twists all equivalent faces.=C2=A0 Only in this ca=
se it's not tiling, but 'fractaling' :D.=C2=A0 I'll have to=
think about that a bit more later!=C2=A0 Seems like it would be one heck o=
f a difficult puzzle anyway!


Chris

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I actually found a good link that shows the coloring I was talking about:
http://www.mathematik.com/Menger/Menger2.html
You can see that even for the smaller void cubes, the insides of the cubies
are also colored based on which direction they are facing.

2010/3/10 Chris Locke

> I've attached an image for clarity.
>
> In the case I was talking of, a single twist won't affect other parts of
> the puzzle, just the part that is twisted. So it's like a normal puzzle,
> just with a crazy shape. In the image I attached, the green outlines
> examples of simple void cube twists that are permitted, and the teal
> outlines the larger twists that are permitted. Basically, the twisting
> works just like you expect it to if a physical level 2 Menger rubik cube
> existed. The mini level 1 cubes can be twisted just like a void cube alo=
ng
> those axes, but in the other directions you have to move a larger 9x9x1 s=
lab
> instead of just a 3x3x1 slice. This means that for the void cubes in the
> corners of the level 2 Menger cube, there is no way to just twist the cub=
ies
> in that individual mini void cube; all twists are twists of 9x9x1 slabs.
> This process can be generalized up to any level as well. So for level 3
> Menger cubes, 3x3x1, 9x9x1, and 27x27x1 twists are possible.
>
> As for sticker coloring, if you look at the level 2 Menger cube, you will
> notice that the smaller 3x3x3 void cubes act as single cubies at that
> level. So if you would color the larger scale level 2 Menger cube simply
> like a void cube, the entire inside faces would be uncolored. In the
> picture I attached, I blacked out the top face of the front-bottom-middle
> void cube to show what that would look like. The only choice would be to
> color that blackened face blue, corresponding to the color directly above
> it.
>
> You're definition of twisting where one twist propagates seems really
> interesting, but hard to visualize :D. Sounds like it could make for a
> pretty interesting puzzle. I'd imagine it might be similar to how in
> Roice's new MagicTile program the infinite tilings are repeated so a twis=
t
> of one face twists all equivalent faces. Only in this case it's not tili=
ng,
> but 'fractaling' :D. I'll have to think about that a bit more later! Se=
ems
> like it would be one heck of a difficult puzzle anyway!
>
> Chris
>
> 2010/3/10 Melinda Green
>
>
>>
>> We may not be talking about the same basic geometry, but if we are then
>> I think that I see the problem. When I say that a twist on a level 1
>> cube (I.E. a Void cube) will affect all other cubes, even of larger
>> levels, meant for twists on larger level cubes to be in proportion to
>> those cubes. So in your 2-level version, twisting a 3x3x1 slice will not
>> only make the same twist on all 19 other level 1 cubes, but will also
>> twist the corresponding 9x9x3 face of the level 2 cube. Rather than
>> mixing up the puzzle beyond all hope (for say a 4 level cube), this
>> design mixes them not at all and seems to leave us with a single void
>> cube to solve.
>>
>> Regarding realizations of possible puzzles, I only mean computer
>> implementations.
>>
>> Regarding the coloring of inside stickers, I didn't completely follow
>> what you proposed, but it doesn't seem like an important issue because I
>> don't think that the inside stickers can contribute to the puzzle. I
>> think they should automatically be correct when the rest of the puzzle
>> is complete. I'd therefore probably color them with the same color as
>> the outside stickers that face in the same direction. That way when the
>> whole puzzle is solved it will appear to be more "complete", but maybe
>> that was what you were saying.
>>
>> So how to define a Rubik's Menger Cube that's harder than a Void cube
>> yet not impossible? Here's another idea: What if a twist on any level 1
>> cube twists all other cubes of all levels that are currently playing the
>> same role? IOW, if you twists a level-1 cube that is part of the FUR
>> cubie, then all other cubes that are also FUR elements of other cubes
>> will twist, and *only* those cubes. That might make all cubes into
>> individually solvable void cubes yet hopelessly interconnected with each
>> other. I don't know but I sense that there may be a natural, elegant,
>> and hard-but-not-impossible-puzzle definition lurking here somewhere.
>>
>> -Melinda
>>
>>
>> Chris Locke wrote:
>> >
>> >
>> > Well, since the void cube is a level 1 Menger sponge, it would always
>> > be possible to go simply to just level 2 for now and have a finite
>> > puzzle in that sense. I don't know if it would be possible to
>> > construct a physical one without having it rounded in shape like the
>> > V-Cubes, but on a computer it would be no problem. How would it
>> > work? Let's say you take the front upper middle edge cube (almost
>> > equivalent to a void cube). You would be able to do L, M, and R
>> > twists on this as much as you want, but if you try, say, a U twist,
>> > you will end up twisting the entire upper layer. This would move the
>> > top 8 cubies of the front upper middle cube and move it to the top 8
>> > cubies of one of the side upper middle cubes, and similarly for other
>> > twists like it. This would give the cube a crazy level of
>> > interconnectedness between all the mini void cubes and would make for
>> > one crazy puzzle to solve. This method could also be extended to any
>> > level of depth desired, except beyond level 2 the number of cubies
>> > just becomes waaay too big for any sane person to deal with :D
>> >
>> > As for coloring of the inside stickers (the void cube is black inside
>> > but the Menger sponge has to have inside colors at least above level 1
>> > but it's no more difficult to have for all cubies) just have the same
>> > color as the whole cube. So you would still have 6 colors, but if you
>> > look at one of the edge pieces of a mini void cube, instead of being 2
>> > colored, it would be 4 colored. But because of the way the coloring
>> > works, the 4 coloredness of the cubie is equivalent to the normal 2
>> > colored edge pieces.
>> >
>> > Chris
>> >
>> > 2010/3/9 Melinda Green l.com>
>> > >>
>>
>> >
>> >
>> >
>> > David,
>> >
>> > I'm aware of Menger cubes, Serpenski gaskets, Cantor dust, etc. Their
>> > constructions are quite simple. Normally quite boringly simple, but I
>> > find this coloration of a Menger cube really quite evocative. I don't
>> > have any more context for the image. I just get Google alerts on the
>> > term "Buddhabrot" which lately includes a lot of postings such as
>> > this
>> > one by a person who used that term as their account name. Quite
>> > flattering really and I often like what they come up with. Yes, the
>> > image appears to be a random Rubik colored Menger cube and not any
>> > sort
>> > of actual puzzle. It just makes me wonder how it could best be
>> > made real.
>> >
>> > My current thought is to treat every 20-cube figure identically,
>> > regardless of scale. So a twist on a 20 "atom" cube would cause a
>> > twist
>> > of every other 20 atom cube as well as every 400 atom cube and so
>> > on up
>> > the chain to some maximum scale cube. Scrambling such a beast
>> > would mix
>> > the colors so completely that at high levels it will just appear as a
>> > single grainy mix of all the colors. You'd need to zoom in to the
>> > atomic
>> > level in order to work on it. In some ways, the fundamental puzzle
>> > would
>> > not be terribly interesting because I think that it would just be
>> > normal
>> > void cube (http://en.wikipedia.org/wiki/Void_Cube) because as you
>> > point
>> > out, the cubies would not mix outside their respective cubes at any
>> > level. Still, it'd be fascinating to watch it being solved.
>> >
>> > Does anyone else have ideas for better ways to make this thing
>> > real? The
>> > only constraint that I urge is that a twist on any scale should have
>> > identical affects on all scales, but just how that might work is
>> > an open
>> > question.
>> >
>> > -Melinda
>> >
>> >
>> >
>> > David Vanderschel wrote:
>> > > This is just a typical fractal generation with a highly
>> > > regular algorithm. (I am trying to distinguish it from the
>> > > more interesting fractals (think of coastlines) that exhibit
>> > > randomness.) Instead of going smaller on each iteration,
>> > > the pattern becomes larger in this case. The basic starting
>> > > pattern is a pile of 20 cubes, corresponding to a 3x3x3
>> > > stack with the central 'cross' (7 cubes) removed. That
>> > > stack, with the holes in it, can be treated as a cube
>> > > itself. So 20 such cubical piles can be piled together in
>> > > the analogous fashion to create the next generation - a pile
>> > > of 400 little cubes. Etc.
>> > >
>> > > The coloring looks random to me. (I can imagine interesting
>> > > looking non-random colorings, some of which could improve
>> > > one's ability to see the picture correctly.)
>> > >
>> > > As an abstract thing, the 20-cube pile could be 'worked'
>> > > like a 3D puzzle. (I have a recollection that there is a
>> > > commercial physical version of such a puzzle.) The only
>> > > catch is that, in the absence of face-center pieces, you
>> > > have to use some other method to assign the face colors.
>> > > However, this is already a familiar problem with the even
>> > > order puzzles. If you use the analogous motions to 'work'
>> > > a 400-cube pile, you see that little cubes can never move
>> > > from one 20-cube pile to another; so that does not lead to
>> > > an especially interesting puzzle. OTOH, the 400-cube
>> > > pile could be regarded as a variation on the order-9
>> > > 3-puzzle; and this one is interesting, as we can see
>> > > cubies that are not in external slices. I.e., we can begin
>> > > to concern ourselves with the permutation and orientation
>> > > of interior cubies that are normally invisible to us.
>> > >
>> > > Melinda, how did you encounter this? Surely there must
>> > > be some context that would provide a little more info about
>> > > its significance (or lack thereof).
>> > >
>> > > I managed to find some context:
>> > > http://dosenjp.tumblr.com/post/430499178/via-dothereject
>> > >
>> > > There is there a comment indicating that this thing has a name -
>> > > a Menger sponge:
>> > > http://en.wikipedia.org/wiki/Menger_sponge
>> > > If you found my explanation of its 'construction' too terse,
>> > > there is a much
>> > > more elaborate version on the wiki.
>> > >
>> > > So the image was basically a coloring of a Menger sponge in the
>> > > manner
>> > > of a Rubik's Cube.
>> > >
>> >
>> >
>> >
>> >
>> >
>>=20=20
>>
>
>

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I actually found a good link that shows the coloring I was talking about:=
=C2=A0 http://www=
.mathematik.com/Menger/Menger2.html
You can see that even for the sm=
aller void cubes, the insides of the cubies are also colored based on which=
direction they are facing.

Melinda wrote:
> I'm aware of Menger cubes, Serpenski gaskets, Cantor dust,
> etc. ...

Yet you had originally written, "I have no idea what's
behind it."; so I figured some explanation would be welcome.

I described the construction as growing at each step because
the presentation makes a point of coloring the lowest level
cubes. For a 'real' sponge, taken to the limit, there are
no visible 'cubies' to color.

> Does anyone else have ideas for better ways to make this
> thing real?

Did you see my suggestion to treat it as a variation on the
order-9 3-puzzle? I meant that in the sense of allowing
twists on the slices that have holes in them. (No problem
for a simulation.) Chris had a slightly different take on
an order-9 variation.

> Regarding the coloring of inside stickers, I didn't
> completely follow what you [Chris] proposed, but it
> doesn't seem like an important issue because I don't think
> that the inside stickers can contribute to the puzzle. I
> think they should automatically be correct when the rest
> of the puzzle is complete.

This is not obvious to me. Do you have proof, reasoning,
reference? I thought this was the interesting thing about
the order-9 variation.

> The only constraint that I urge is that a twist on any scale
> should have
> identical affects on all scales, but just how that might work
> is an open
> question.

I suppose something along those lines is what it would take
to call the puzzle "fractal". But I can't think of any
useful way to make it work either.

Speaking of "fractal", it should be noted that these various
space-filling curves were known before Mandelbrot appeared
on the scene. They are so deliberately self-similar that I
have never regarded them as particularly interesting in the
larger fractal context.

In that coloring that Chris pointed us to, it may noted that
it is equivalent to a pile of identically decorated cubies
in which every cubie has stickers on all six faces with the
starting position having the same color facing in each of
the six possible directions. Interestingly, even the basic
order-3 3-puzzle can be viewed this way. With this view,
what makes the cubies distinguishable is that the _visible_
set of stickers on each cubie is unique and unchanging.

Regards,
David V.

Chris & David,

Both of your suggestions allow for twists on elements such as a 9x9x1
slice which are not part of any sort of standard cube of any scale.
Also, twists in both of your designs have only local effects. I don't
want to put them down because I'm actually very happy to see some folks
giving this problem some thought. While we might come up with a hard yet
solvable puzzle this way, these aspects just don't get me excited. As
you both point out, these purely local operations are not very
fractal-like and therefore don't exploit the fundamental nature of this
geometry.

Yes, I was thinking of interactions very much like Roice's Magic Tile in
which portions of the geometry are not just copies of other portions but
in actual fact *are* the same bits of geometry. I was hoping to do the
same thing but with in scale as well as in position, but maybe that's
not the best approach. At least it didn't seem to be generating anything
terribly new.

Well here is yet another possible design that I would like to offer: A
click on a given sticker will affect all twistable 3^n x 3^n x 3^(n-1)
faces that contain that sticker. IOW, clicking on a sticker will twist
the face of its level 1 void cube as well as the level 2 cube that
contains it, and then level 3 cube that contains that one, and so on up
to some maximum (probably ending right there). If we don't allow twists
on "inner" stickers, then not all faces in every such chain will be
twistable. I'm fine with whichever design makes more sense. This gives a
puzzle that is sort of a combination of local and fractal.

Thoughts?
-Melinda

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P.S. One more thought and a correction:
The twisting in the fractal actually involves clipping of the faces. Like,
in the level 2 case twisting the front of a 20 cubie atom will result in the
3x3x1 cubies to clip into their neighbors. Not actually a problem for
software, but it means that my statement that the fractal twisting can be
implemented with my proposed physical twisting is not correct because the
fractal twisting has some seemingly fundamentally impossible to physically
implement twisting going on.
Chris

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P.S.=C2=A0 One more thought and a correction:
The twisting in the fracta=
l actually involves clipping of the faces.=C2=A0 Like, in the level 2 case =
twisting the front of a 20 cubie atom will result in the 3x3x1 cubies to cl=
ip into their neighbors.=C2=A0 Not actually a problem for software, but it =
means that my statement that the fractal twisting can be implemented with m=
y proposed physical twisting is not correct because the fractal twisting ha=
s some seemingly fundamentally impossible to physically implement twisting =
going on.

Chris


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I've had a bit more time to think about the fractal-twisting idea and here
are my thoughts on it (and please let me know if I'm misrepresenting your
ideas here, Melinda):

Since if you have a single twist make a corresponding 3x3x1, twist, a 9x9x3
twist, a 27x27x9 twist, and so on, the twisting propagates up the Menger
levels all the way to the top level. So if you have a level 1 puzzle, you
will have the sticker you clicked twist in one way; if you have a level 2
puzzle it will twist in two ways, and so on. In this way, the puzzle is
seems like the level you work with will play a fundamental role in the way
twisting works. This isn't necessarily a bad thing, just an observation.
Now, Melinda has proposed what seem like two potential ways that twisting
will work in this fashion. The first where a twist on a single 20 cube
"atom" will also twist every other 20 cube "atom", and the equivalent twist=
s
will also apply to each 400 cube atom, and so on. This will mean that each
twist is truly global in every sense of the word. As for how the puzzle
will actually function, my intuition from playing with some of the highly
interconnected puzzles in MagicTile, is that due to the globalness of the
moves and the potential size of the puzzle, would make for one of the most
mind-bendingly difficult puzzles to solve. There's also the more distant
(but the more I think about it increasing) present possibility that this
high level of interconnected-ness might reduce the puzzle in complexity
(like it might collapse into a single void cube). Without a working model
to work with, I can't say for sure.

The second form of twisting is a play on the same theme, but with a little
bit less everything-is-twisting global-ness. In this case the sticker you
click affects which 3^n x 3^n x 3^(n-1) twists are done. So for instance,
on the level 2 puzzle, clicking one of the front face stickers won't cause
every mini 20 cube atom to twist, but just the one containing the sticker
you clicked. It will also cause the entire front 9x9x3 face to twist. So
if you left-clicked a sticker on the front-upper-middle atom, then that
3x3x1 face will undergo a counter-clockwise twist, and the twisting of the
whole front 9x9x3 will cause that atom to also itself move to the
front-left-middle atom's spot. If the puzzle was level 3, then the atom yo=
u
clicked would twist, move within the level 2 atom, and then move within the
level 3 atom as well for a total of 3 moves. Now, with this case of
twisting, you still get some globalness to each twist, but because you
aren't affecting essentially the whole puzzle at once, I believe that the
possibility it would reduce to a single void cube in terms of complexity
vanishes. This method of twisting I could see becoming a standard if this
puzzle was implemented. Although I think for human reasons the size might
be limited to level 3 (although I've been surprised with what some people
have conquered in the past.... 7^5 and 120-cell anyone? :D).

Now, I think that the method of twisting I proposed would have some merit a=
s
well. It might not be exploiting the fractalness of the shape, but it does
exploit the weirdness of the shape by taking the obvious physical
interpretation of what twisting would do. Furthermore, because the 9x9x1
twist moves around pieces that can be moved by 3x3x1 twists also, there is
still a large amount of global mixing and inter-connectedness going on. In
a sense, actually, this method of twisting is the most generalizable becaus=
e
the fractal twists can be imitated through selective choices of the twists
done here. For the level 2 puzzle, you could implement the second method o=
f
fractal twisting by doing a single 3x3x1 twist, then three 9x9x1 twists.
Naturally, though, there is a certain elegance to the fractal twisting that
makes it quite appealing (and potentially ground-breaking in the
puzzle-scene), but I think both kinds of twisting would result in
interesting puzzles. My proposed example might be more mundane in it's
definition, but by very nature of the geometry in question it becomes quite
interesting in its own right. I guess it's like the new additionally
puzzles of MagicCube4D in the sense that there's no new deep innovation in
them, but they draw their intrigue from their interesting shapes, which
themselves give rise to new and unseen difficulties.

..... I wonder if there's a way we can make this idea a reality ^_^

Chris

2010/3/11 Melinda Green

>
>
> Chris & David,
>
> Both of your suggestions allow for twists on elements such as a 9x9x1
> slice which are not part of any sort of standard cube of any scale.
> Also, twists in both of your designs have only local effects. I don't
> want to put them down because I'm actually very happy to see some folks
> giving this problem some thought. While we might come up with a hard yet
> solvable puzzle this way, these aspects just don't get me excited. As
> you both point out, these purely local operations are not very
> fractal-like and therefore don't exploit the fundamental nature of this
> geometry.
>
> Yes, I was thinking of interactions very much like Roice's Magic Tile in
> which portions of the geometry are not just copies of other portions but
> in actual fact *are* the same bits of geometry. I was hoping to do the
> same thing but with in scale as well as in position, but maybe that's
> not the best approach. At least it didn't seem to be generating anything
> terribly new.
>
> Well here is yet another possible design that I would like to offer: A
> click on a given sticker will affect all twistable 3^n x 3^n x 3^(n-1)
> faces that contain that sticker. IOW, clicking on a sticker will twist
> the face of its level 1 void cube as well as the level 2 cube that
> contains it, and then level 3 cube that contains that one, and so on up
> to some maximum (probably ending right there). If we don't allow twists
> on "inner" stickers, then not all faces in every such chain will be
> twistable. I'm fine with whichever design makes more sense. This gives a
> puzzle that is sort of a combination of local and fractal.
>
> Thoughts?
> -Melinda
>=20=20
>

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I've had a bit more time to think about the fractal-twisting idea and h=
ere are my thoughts on it (and please let me know if I'm misrepresentin=
g your ideas here, Melinda):

Since if you have a single twist make a=
corresponding 3x3x1, twist, a 9x9x3 twist, a 27x27x9 twist, and so on, the=
twisting propagates up the Menger levels all the way to the top level.=C2=
=A0 So if you have a level 1 puzzle, you will have the sticker you clicked =
twist in one way; if you have a level 2 puzzle it will twist in two ways, a=
nd so on.=C2=A0 In this way, the puzzle is seems like the level you work wi=
th will play a fundamental role in the way twisting works.=C2=A0 This isn&#=
39;t necessarily a bad thing, just an observation.=C2=A0 Now, Melinda has p=
roposed what seem like two potential ways that twisting will work in this f=
ashion.=C2=A0 The first where a twist on a single 20 cube "atom" =
will also twist every other 20 cube "atom", and the equivalent tw=
ists will also apply to each 400 cube atom, and so on.=C2=A0 This will mean=
that each twist is truly global in every sense of the word.=C2=A0 As for h=
ow the puzzle will actually function, my intuition from playing with some o=
f the highly interconnected puzzles in MagicTile, is that due to the global=
ness of the moves and the potential size of the puzzle, would make for one =
of the most mind-bendingly difficult puzzles to solve.=C2=A0 There's al=
so the more distant (but the more I think about it increasing) present poss=
ibility that this high level of interconnected-ness might reduce the puzzle=
in complexity (like it might collapse into a single void cube).=C2=A0 With=
out a working model to work with, I can't say for sure.


The second form of twisting is a play on the same theme, but with a lit=
tle bit less everything-is-twisting global-ness.=C2=A0 In this case the sti=
cker you click affects which 3^n x 3^n x 3^(n-1) twists are done.=C2=A0 So =
for instance, on the level 2 puzzle, clicking one of the front face sticker=
s won't cause every mini 20 cube atom to twist, but just the one contai=
ning the sticker you clicked.=C2=A0 It will also cause the entire front 9x9=
x3 face to twist.=C2=A0 So if you left-clicked a sticker on the front-upper=
-middle atom, then that 3x3x1 face will undergo a counter-clockwise twist, =
and the twisting of the whole front 9x9x3 will cause that atom to also itse=
lf move to the front-left-middle atom's spot.=C2=A0 If the puzzle was l=
evel 3, then the atom you clicked would twist, move within the level 2 atom=
, and then move within the level 3 atom as well for a total of 3 moves.=C2=
=A0 Now, with this case of twisting, you still get some globalness to each =
twist, but because you aren't affecting essentially the whole puzzle at=
once, I believe that the possibility it would reduce to a single void cube=
in terms of complexity vanishes.=C2=A0 This method of twisting I could see=
becoming a standard if this puzzle was implemented.=C2=A0 Although I think=
for human reasons the size might be limited to level 3 (although I've =
been surprised with what some people have conquered in the past.... 7^5 and=
120-cell anyone? :D).


Now, I think that the method of twisting I proposed would have some mer=
it as well.=C2=A0 It might not be exploiting the fractalness of the shape, =
but it does exploit the weirdness of the shape by taking the obvious physic=
al interpretation of what twisting would do.=C2=A0 Furthermore, because the=
9x9x1 twist moves around pieces that can be moved by 3x3x1 twists also, th=
ere is still a large amount of global mixing and inter-connectedness going =
on.=C2=A0 In a sense, actually, this method of twisting is the most general=
izable because the fractal twists can be imitated through selective choices=
of the twists done here.=C2=A0 For the level 2 puzzle, you could implement=
the second method of fractal twisting by doing a single 3x3x1 twist, then =
three 9x9x1 twists.=C2=A0 Naturally, though, there is a certain elegance to=
the fractal twisting that makes it quite appealing (and potentially ground=
-breaking in the puzzle-scene), but I think both kinds of twisting would re=
sult in interesting puzzles.=C2=A0 My proposed example might be more mundan=
e in it's definition, but by very nature of the geometry in question it=
becomes quite interesting in its own right.=C2=A0 I guess it's like th=
e new additionally puzzles of MagicCube4D in the sense that there's no =
new deep innovation in them, but they draw their intrigue from their intere=
sting shapes, which themselves give rise to new and unseen difficulties.>

..... I wonder if there's a way we can make this idea a reality ^_^=


Chris

Melinda, relative to my previous post in this thread (which was
primarily a response to your preceding post), you failed to
answer a direct question for which I was eager to see your
answer. So I will repeat the question.

You had written:
>> Regarding the coloring of inside stickers, I didn't
>> completely follow what you [Chris] proposed, but it
>> doesn't seem like an important issue because I don't think
>> that the inside stickers can contribute to the puzzle. I
>> think they should automatically be correct when the rest
>> of the puzzle is complete.

and I had asked:
>This is not obvious to me. Do you have proof, reasoning,
>reference? I thought this was the interesting thing about
>the order-9 variation.

Regards,
David V.

David,

Indeed I failed. I do that a lot. ;-)

I didn't have a proof of my guess, but that's what my "I think"
equivocation meant: It was just my guess. It seems entirely clear to me
that inner stickers cannot mix with outer stickers given the twists
we've talked about so far. I'll go even further and say that I'm pretty
sure that inner stickers can never be adjacent to any outer stickers
that are not on its own cubie. As to a proof? Well, let's see. Level-1
cubies with inner stickers will always have two inner and two outer
stickers arranged in a ring: outer, outer, inner, inner. Given a uniform
pristine coloring like we've been considering, every level-1 edge cubie
with outer colors O1,O2 will have inner colors I1,I2 that are identical
to every other cubie with the same pair of outer colors. Since all edge
cubies with the same pair of outer stickers are identical and there is
only one correct orientation for any edge piece, it seems to follow that
once all of the outer stickers are placed correctly, all inner stickers
will also have to also be correct.

-Melinda

David Vanderschel wrote:
> Melinda, relative to my previous post in this thread (which was
> primarily a response to your preceding post), you failed to
> answer a direct question for which I was eager to see your
> answer. So I will repeat the question.
>
> You had written:
>
>>> Regarding the coloring of inside stickers, I didn't
>>> completely follow what you [Chris] proposed, but it
>>> doesn't seem like an important issue because I don't think
>>> that the inside stickers can contribute to the puzzle. I
>>> think they should automatically be correct when the rest
>>> of the puzzle is complete.
>>>
>
> and I had asked:
>
>> This is not obvious to me. Do you have proof, reasoning,
>> reference? I thought this was the interesting thing about
>> the order-9 variation.
>>
>
> Regards,
> David V.
>
>

As far as I can tell, Melinda is only talking about stickers
on cubies that are in external slices. Yes, the previously
hidden stickers on those cubies, though now visible, do
remain 'inside' (i.e., never facing outwards); and their
orientation is determined by that of the stickers which do
face outwards.

What is interesting about the order-9-with-Menger-holes is
that you can now see cubies that are not in _any_ external
slice, making it possible to see what is happening to some
permutations and orientations of cubies down in the interior
of the puzzle. The stickers on these interior cubies are
also stickers that I would consider to be "inside stickers".
What I was questioning is whether these deeply interior
stickers (indeed, the interior cubies themselves) will
necessarily be correct when the outer stickers are
correct. That is still not obvious to me, and I am inclined
to doubt that it holds. For me, that makes this variation
interesting in a new sense that we have not previously
concerned ourselves with.

Regards,
David V.


----- Original Message -----
From: "Melinda Green"
To: <4D_Cubing@yahoogroups.com>
Sent: Thursday, March 11, 2010 8:43 PM
Subject: Re: [MC4D] Fractal cubes


David,

Indeed I failed. I do that a lot. ;-)

I didn't have a proof of my guess, but that's what my "I think"
equivocation meant: It was just my guess. It seems entirely
clear to me
that inner stickers cannot mix with outer stickers given the
twists
we've talked about so far. I'll go even further and say that I'm
pretty
sure that inner stickers can never be adjacent to any outer
stickers
that are not on its own cubie. As to a proof? Well, let's see.
Level-1
cubies with inner stickers will always have two inner and two
outer
stickers arranged in a ring: outer, outer, inner, inner. Given a
uniform
pristine coloring like we've been considering, every level-1
edge cubie
with outer colors O1,O2 will have inner colors I1,I2 that are
identical
to every other cubie with the same pair of outer colors. Since
all edge
cubies with the same pair of outer stickers are identical and
there is
only one correct orientation for any edge piece, it seems to
follow that
once all of the outer stickers are placed correctly, all inner
stickers
will also have to also be correct.

-Melinda

David Vanderschel wrote:
> Melinda, relative to my previous post in this thread (which was
> primarily a response to your preceding post), you failed to
> answer a direct question for which I was eager to see your
> answer. So I will repeat the question.
>
> You had written:
>
>>> Regarding the coloring of inside stickers, I didn't
>>> completely follow what you [Chris] proposed, but it
>>> doesn't seem like an important issue because I don't think
>>> that the inside stickers can contribute to the puzzle. I
>>> think they should automatically be correct when the rest
>>> of the puzzle is complete.
>>>
>
> and I had asked:
>
>> This is not obvious to me. Do you have proof, reasoning,
>> reference? I thought this was the interesting thing about
>> the order-9 variation.
>>
>
> Regards,
> David V.

Hi guys,
I'm new to this very interesting group. Last couple of days I tried to fo=
llow the discussion, but maybe I've missed something, so I may say somethin=
g that is already known.=20
As for this fractal puzzle, first thing is that you need stickers of the =
external surface of any 3x3x3 block. Some of stickers will be "very deep in=
side" the original model, but they will be visible after the first turn - w=
hen we rotate 3x3x1 slice of the corner block, we'll see some deepinside st=
ickers if they'll not go back inside because of rotation of 9x9x3 block and=
27x27x9 block (but in that case other "deep inside" stickers will go back =
to the surface).
For the start we may paint every block in 6 colors (all in the same orien=
tation). So it will be like solving of 20 (or 400) void cubes by the same s=
equence of operations, but some of these cubes will change their orientatio=
n after each move, so pictures on the blocks will be different even after t=
he second twist.
It may be possible that in the solved puzzle some of 3x3x3 blocks will be=
not in their original positions. But it should not be a problem: puzzle (e=
ven level 2 - of 20 blocks) seems to be very difficult.