To: moz-do-not-send=3D"true" title=3D"4D_Cubing@yahoogroups.com"
href=3D"mailto:4D_Cubing@yahoogroups.com">MagicCube4D >
Sent: Thursday, February
09, 2017 6:38 AM
Subject: [MC4D] Physical 4D
puzzle achieved
=C2=A0 Dear cubists,
I am excited to announce that I've been able to implement a
working 2^4 in the real world! It uses moz-do-not-send=3D"true"
href=3D"https://groups.yahoo.com/neo/groups/4D_Cubing/conversations/message=
s/3207">a
design I had described a little over a year ago. The
key was a clever arrangement of magnets=C2=A0 that the Matholog=
er
came up with for implementing a 2^3 puzzle. I realized that
it could be extended to give me a mechanism to support my
design. I then prototyped it and it works quite well. Here
is a quick video I made just for you:
=C2=A0=C2=A0=C2=A0 =C2=A0=C2=A0=C2=A0 e"
class=3D"moz-txt-link-freetext"
href=3D"https://www.youtube.com/watch?v=3DAsx653BGDWA">https:=
//www.youtube.com/watch?v=3DAsx653BGDWA
Please do not share this link because it is unpublished and
I intend to delete and replace it with a proper video. It is
mainly meant for getting your feedback and advice. Please
reply here, or privately to class=3D"moz-txt-link-abbreviated"
href=3D"mailto:melinda@superliminal.com">melinda@superliminal=
.com
if you're shy. Feel free to comment on all aspects of the
video, and not just on the mathematics. For instance, feel
free to comment on the video and audio quality, or anything
else that strikes you.
Many of us have dreamed about such a puzzle for a long time
and it's become something of a holy grail. It's always
seemed almost impossible to create any sort of 4D puzzle in
the real world, but there was never any clear reason why it
couldn't be done. The real holy grail would be a physical
3^4, so maybe this is just the baby grail. Of course maybe
this design can be extended to implement a 3^4. What do you
think?
Happy puzzling!
-Melinda
=20=20=20=20=20=20=20=20
--------------D860B8DBA114081DF8A9726C--
From: liamjwright@btinternet.com
Date: 09 Feb 2017 23:05:38 +0000
Subject: Re: [MC4D] Physical 4D puzzle achieved
From: liamjwright@btinternet.com
Date: 09 Feb 2017 23:19:19 +0000
Subject: Re: [MC4D] Physical 4D puzzle achieved
From: damienturtle@hotmail.co.uk
Date: Thu, 9 Feb 2017 15:37:37 -0800
Subject: Re: Physical 4D puzzle achieved
--------------30D90E8823CB253FED57C2B9
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Hello Liam,
Yes, a new perspective can sometimes change everything. In this case, I wouldn't guess that this version will be easier to solve or understand than a virtual version, but there is undoubtedly something satisfying about puzzles you can hold in your hands. One way it may be helpful is in showing the difference between stickers and pieces. The physical pieces are now obvious, though the stickers are no longer cubes, so that is a little misleading. It may make for a gateway puzzle that gets some people to look more deeply into the virtual puzzles. I'll be curious to see how people in the general puzzle community react when I show it around.
And yes, the magnet restrictions definitely seems like a kind of bandaging. Extending the Mathologer's arrangement allows two main implementations that will support the basic moves. (Mathologer video for reference. ) I chose the one that seems to allow for more unusual moves such as the restacking move, but the other version might allow for some other legal, breakthrough move. Maybe I'll just need to prototype that one too.
Thanks!
-Melinda
On 2/9/2017 8:38 AM, liamjwright@btinternet.com [4D_Cubing] wrote:
> Congratulations!
>
> If there's anything I've learned exploring different puzzles, it's that a different perspective can totally change the difficulty of a puzzle. It would definitely be interesting to see if this version of a 2^4 makes it any easier or harder to solve/understand it.
>
> When you split the cube down the middle, whether it produces a legal permutation of the cube or not, I think it definitely has the potential to create a slightly odd new puzzle. The way the magnets limit rotation after the split could perhaps act in the same way that a bandage does on a regular 3D puzzle?
>
> -Liam
--------------30D90E8823CB253FED57C2B9
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable
">
Hello Liam,
Yes, a new perspective can sometimes change everything. In this
case, I wouldn't guess that this version will be easier to solve or
understand than a virtual version, but there is undoubtedly
something satisfying about puzzles you can hold in your hands. One
way it may be helpful is in showing the difference between stickers
and pieces. The physical pieces are now obvious, though the stickers
are no longer cubes, so that is a little misleading. It may make for
a gateway puzzle that gets some people to look more deeply into the
virtual puzzles. I'll be curious to see how people in the general
puzzle community react when I show it around.
And yes, the magnet restrictions definitely seems like a kind of
bandaging. Extending the Mathologer's arrangement allows two main
implementations that will support the basic moves. ( href=3D"https://www.youtube.com/watch?v=3DXb8ENlS-5Go">Mathologer
video for reference.) I chose the one that seems to allow for
more unusual moves such as the restacking move, but the other
version might allow for some other legal, breakthrough move. Maybe
I'll just need to prototype that one too.
Thanks!
-Melinda
=C2=A0
Congratulations!
If there's anything I've learned exploring different puzzles,
it's that a different perspective can totally change the
difficulty of a puzzle. It would definitely be interesting to
see if this version of a 2^4 makes it any easier or harder to
solve/understand it.
When you split the cube down the middle, whether it produces
a legal permutation of the cube or not, I think it definitely
has the potential to create a slightly odd new puzzle. The way
the magnets limit rotation after the split could perhaps act in
the same way that a bandage does on a regular 3D puzzle?
-Liam
--------------30D90E8823CB253FED57C2B9--
From: Christopher Locke <project.eutopia@gmail.com>
Date: Thu, 9 Feb 2017 16:09:36 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--------------260FDF0821C1A804CDD654D0
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Content-Transfer-Encoding: 8bit
Hello Melinda,
Congratulations! Looks very nice.
I noticed when looking at your puzzle, that the rough correspondence is
the following:
- Each left and right 2x2x2 block corresponds to the +/- w direction
faces of the 4D cube, with the respective stickers on the 8 corners of
each block. So the w-direction stickers are initially on the outer
corners. For reference, let's call the left - and right +
- By arbitrarily setting the central 8 stickers that bridge the gap
between the two halves to be the -y face, then the +y face is split
between the left and right open faces
- The stickers in the top layer that are facing upwards are the split +z
face stickers, and those in the bottom are the split -z face stickers
(can be brought together by rotation).
- Similarly, for the front (+x) and back (-x) faces
- The faces seem to live as follows: (-w leftmost 8 blocks, +w rightmost
8 blocks, +z topmost 8 blocks, -z bottommost 8 blocks, +x frontmost 8
blocks, -x backmost 8 blocks, +y middle 8 blocks, -y leftmost 4 and
rightmost 4 blocks).
- Stickers are as follows: (-w are on outer corners of leftmost 8
blocks, +w on outer corners of leftmost 8 blocks, +z in the middle of
the top face of the left and right 2x2x2 blocks, and -z in the middle of
the bottom face, +x in the middle of the front face of the left and
right 2x2x2 blocks, and -x in the middle of the back facing part, +y the
covered up middle 8 stickers, and -y the leftmost and rightmost middle
stickers.
- From this point of view, comparing to the 4D model in MagicCube4D
(where the -w face is in the center, and the +/- x,y,z faces are
surrounding it, with +w hidden), then 90 degree twists of the -w face
are achieved by rotating the left block arbitrarily, and 90 degree
twists of the +w face by rotating the right block arbitrarily.
- From this point of view, your rotations of both faces
simultaneously are two twists of the inner and outermost halves,
resulting in a 3D rotation of the MagicCube4D model that keeps the
central -w face in the center (plain 3D rotations)
- Similarly, it seems that 180 degree rotations of, say, the topmost +z
face, seems to be a 180 degree rotation of the topmost +z face in
MagicCube4D around the y-axis (keeps the +/-y stickers on the +/-y face)
- 180 degree rotation of front block is a rotation of +x face around the
y-axis
- Note that although you only have 180 degree twists about the
y-axis, a rotation of the puzzle can bring it into an orientation to
probably do any 180 degree twist of the x, y, or z faces.
The problem here is that the outer +/-w stickers always stay on the 8
corners of the left and right 2x2x2 blocks. This means that you cannot
mix up w stickers with x, y, or z stickers yet, without doing some
inversion of the small 3D cubes.
In conclusion, without yet considering the "inversion" type moves, you
have all 90 degree twists of the w faces, and 180 degree twists of the
x, y, z faces. The trick is to find out how to legally mix the w
stickers up with x, y, z stickers. If you can do this, then the 90
degree flexibility of the w face twists should make it possible to reach
any permutation. I will think about this more on the weekend, using
some pen and paper instead of just thinking while walking to work (it is
hard to visualize the inversion moves in my head :D).
Best regards,
Chris
On 2017年02月09日 15:37, Melinda Green melinda@superliminal.com [4D_Cubing]
wrote:
>
> Hello Liam,
>
> Yes, a new perspective can sometimes change everything. In this case,
> I wouldn't guess that this version will be easier to solve or
> understand than a virtual version, but there is undoubtedly something
> satisfying about puzzles you can hold in your hands. One way it may be
> helpful is in showing the difference between stickers and pieces. The
> physical pieces are now obvious, though the stickers are no longer
> cubes, so that is a little misleading. It may make for a gateway
> puzzle that gets some people to look more deeply into the virtual
> puzzles. I'll be curious to see how people in the general puzzle
> community react when I show it around.
>
> And yes, the magnet restrictions definitely seems like a kind of
> bandaging. Extending the Mathologer's arrangement allows two main
> implementations that will support the basic moves. (Mathologer video
> for reference. ) I chose
> the one that seems to allow for more unusual moves such as the
> restacking move, but the other version might allow for some other
> legal, breakthrough move. Maybe I'll just need to prototype that one too.
>
> Thanks!
> -Melinda
>
> On 2/9/2017 8:38 AM, liamjwright@btinternet.com [4D_Cubing] wrote:
>> Congratulations!
>>
>> If there's anything I've learned exploring different puzzles, it's
>> that a different perspective can totally change the difficulty of a
>> puzzle. It would definitely be interesting to see if this version of
>> a 2^4 makes it any easier or harder to solve/understand it.
>>
>> When you split the cube down the middle, whether it produces a legal
>> permutation of the cube or not, I think it definitely has the
>> potential to create a slightly odd new puzzle. The way the magnets
>> limit rotation after the split could perhaps act in the same way that
>> a bandage does on a regular 3D puzzle?
>>
>> -Liam
>
>
--------------260FDF0821C1A804CDD654D0
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Hello Melinda,
Congratulations! Looks very nice.
I noticed when looking at your puzzle, that the rough
correspondence is the following:
- Each left and right 2x2x2 block corresponds to the +/- w
direction faces of the 4D cube, with the respective stickers on
the 8 corners of each block. So the w-direction stickers are
initially on the outer corners. For reference, let's call the
left - and right +
- By arbitrarily setting the central 8 stickers that bridge the
gap between the two halves to be the -y face, then the +y face is
split between the left and right open faces
- The stickers in the top layer that are facing upwards are the
split +z face stickers, and those in the bottom are the split -z
face stickers (can be brought together by rotation).
- Similarly, for the front (+x) and back (-x) faces
- The faces seem to live as follows: (-w leftmost 8 blocks, +w
rightmost 8 blocks, +z topmost 8 blocks, -z bottommost 8 blocks,
+x frontmost 8 blocks, -x backmost 8 blocks, +y middle 8 blocks,
-y leftmost 4 and rightmost 4 blocks).
- Stickers are as follows:Â (-w are on outer corners of leftmost
8 blocks, +w on outer corners of leftmost 8 blocks, +z in the
middle of the top face of the left and right 2x2x2 blocks, and -z
in the middle of the bottom face, +x in the middle of the front
face of the left and right 2x2x2 blocks, and -x in the middle of
the back facing part, +y the covered up middle 8 stickers, and -y
the leftmost and rightmost middle stickers.
- From this point of view, comparing to the 4D model in
MagicCube4D (where the -w face is in the center, and the +/- x,y,z
faces are surrounding it, with +w hidden), then 90 degree twists
of the -w face are achieved by rotating the left block
arbitrarily, and 90 degree twists of the +w face by rotating the
right block arbitrarily.
   - From this point of view, your rotations of both faces
simultaneously are two twists of the inner and outermost halves,
resulting in a 3D rotation of the MagicCube4D model that keeps the
central -w face in the center (plain 3D rotations)
- Similarly, it seems that 180 degree rotations of, say, the
topmost +z face, seems to be a 180 degree rotation of the topmost
+z face in MagicCube4D around the y-axis (keeps the +/-y stickers
on the +/-y face)
- 180 degree rotation of front block is a rotation of +x face
around the y-axis
   - Note that although you only have 180 degree twists about
the y-axis, a rotation of the puzzle can bring it into an
orientation to probably do any 180 degree twist of the x, y, or z
faces.
The problem here is that the outer +/-w stickers always stay on
the 8 corners of the left and right 2x2x2 blocks. This means that
you cannot mix up w stickers with x, y, or z stickers yet, without
doing some inversion of the small 3D cubes.
In conclusion, without yet considering the "inversion" type
moves, you have all 90 degree twists of the w faces, and 180
degree twists of the x, y, z faces. The trick is to find out how
to legally mix the w stickers up with x, y, z stickers. If you
can do this, then the 90 degree flexibility of the w face twists
should make it possible to reach any permutation. I will think
about this more on the weekend, using some pen and paper instead
of just thinking while walking to work (it is hard to visualize
the inversion moves in my head :D).
Best regards,
Chris
cite="mid:704f6044-c5d7-447b-d63f-4de386c4c75a@superliminal.com"
type="cite"> Â
Hello Liam,
Yes, a new perspective can sometimes change everything. In
this case, I wouldn't guess that this version will be
easier to solve or understand than a virtual version, but
there is undoubtedly something satisfying about puzzles
you can hold in your hands. One way it may be helpful is
in showing the difference between stickers and pieces. The
physical pieces are now obvious, though the stickers are
no longer cubes, so that is a little misleading. It may
make for a gateway puzzle that gets some people to look
more deeply into the virtual puzzles. I'll be curious to
see how people in the general puzzle community react when
I show it around.
And yes, the magnet restrictions definitely seems like a
kind of bandaging. Extending the Mathologer's arrangement
allows two main implementations that will support the
basic moves. ( href="https://www.youtube.com/watch?v=Xb8ENlS-5Go">Mathologer
video for reference.) I chose the one that seems to
allow for more unusual moves such as the restacking move,
but the other version might allow for some other legal,
breakthrough move. Maybe I'll just need to prototype that
one too.
Thanks!
-Melinda
On 2/9/2017 8:38 AM,
moz-do-not-send="true" class="moz-txt-link-abbreviated"
href="mailto:liamjwright@btinternet.com">liamjwright@btinternet.com
[4D_Cubing] wrote:
type="cite">
 Congratulations!
If there's anything I've learned exploring different
puzzles, it's that a different perspective can totally
change the difficulty of a puzzle. It would definitely
be interesting to see if this version of a 2^4 makes it
any easier or harder to solve/understand it.
When you split the cube down the middle, whether it
produces a legal permutation of the cube or not, I think
it definitely has the potential to create a slightly odd
new puzzle. The way the magnets limit rotation after the
split could perhaps act in the same way that a bandage
does on a regular 3D puzzle?
-Liam
--------------260FDF0821C1A804CDD654D0--
From: Melinda Green <melinda@superliminal.com>
Date: Thu, 9 Feb 2017 16:46:40 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--------------A24FA9F6781064FE4958154F
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Hello Matt,
Great to hear from you again! For the newer members, Matt currently holds the record for the shortest 3^4 solution with 227 twists.
Good question regarding orientations. No, it hasn't come up yet, and I feel a little embarrassed for not considering it more closely in light of the red-blue problem, but then that's what the preview video is for! No, you can't arbitrarily place any piece in any orientation. With this arrangement of magnets, each of the red or blue corners can be placed in any of the 3 orientations that leave it in the outer corner position, but no others. I guess that means the problem is unsolvable with this arrangement of magnets.
If I had used the "other" natural arrangement, it would also be possible to achieve 3 more orientations by turning a red or blue sticker all the way around and sticking it into the center of its 2x2x2 cube. That would be interesting, but would be a completely invalid orientation, leaving a 3-sticker junction sticking out of the corner and with the patterns of adjacent pieces not matching at all. By that I mean that the diagonal lines between colors on a piece would meet adjacent pieces in an 'X' configuration rather than matching diagonals. In other words, you could get a mixing of the the normal view with the crazy inside-out views you see in the video, but I don't think it would be helpful. Those 3 inverted piece orientations are not among the 12 tetrahedral orientations that you are talking about however. They are from the additional 12 orientations of the cubic pieces, but it is an interesting aside.
Thanks Matt!
-Melinda
On 2/9/2017 3:19 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>
>
> Very impressive Melinda!
>
> Just a quick reply for now, I'll try to find time to think about this in more detail. I may have missed this in the video, so apologies if this has been addressed already, but have you checked that each piece works in each of the 12 possible orientations with the arrangement of magnets? I'm hoping there's some simple way to allow the puzzle to fully scramble.
>
> Matt
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Hello Matt,
Great to hear from you again! For the newer members, Matt currently
holds the record for the shortest 3^4 solution with 227 twists.
Good question regarding orientations. No, it hasn't come up yet, and
I feel a little embarrassed for not considering it more closely in
light of the red-blue problem, but then that's what the preview
video is for! No, you can't arbitrarily place any piece in any
orientation. With this arrangement of magnets, each of the red or
blue corners can be placed in any of the 3 orientations that leave
it in the outer corner position, but no others. I guess that means
the problem is unsolvable with this arrangement of magnets.
If I had used the "other" natural arrangement, it would also be
possible to achieve 3 more orientations by turning a red or blue
sticker all the way around and sticking it into the center of its
2x2x2 cube. That would be interesting, but would be a completely
invalid orientation, leaving a 3-sticker junction sticking out of
the corner and with the patterns of adjacent pieces not matching at
all. By that I mean that the diagonal lines between colors on a
piece would meet adjacent pieces in an 'X' configuration rather than
matching diagonals. In other words, you could get a mixing of the
the normal view with the crazy inside-out views you see in the
video, but I don't think it would be helpful. Those 3 inverted piece
orientations are not among the 12 tetrahedral orientations that you
are talking about however. They are from the additional 12
orientations of the cubic pieces, but it is an interesting aside.
Thanks Matt!
-Melinda
Very impressive Melinda!
Just a quick reply for now, I'll try to find time to think about
this in more detail. I may have missed this in the video, so
apologies if this has been addressed already, but have you checked
that each piece works in each of the 12 possible orientations with
the arrangement of magnets? I'm hoping there's some simple way to
allow the puzzle to fully scramble.
Matt
--------------A24FA9F6781064FE4958154F--
From: Nan Ma <mananself@gmail.com>
Date: Thu, 9 Feb 2017 17:20:36 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--Apple-Mail-FE53156F-3162-4706-8963-014F1892A6F5
Content-Type: text/plain;
charset=utf-8
Content-Transfer-Encoding: quoted-printable
Hi Melinda,
This is a great achievement!
I haven't convinced myself this implementation covers all the permutations =
of 2^4. Maybe someone can create a GAP script to quickly verify the size of=
the group as a sanity check.
In order to illustrate the idea, can you create a 2D version of 2^3 in the =
same way, using eight square tiles?
Nan
Sent from my iPhone
> On Feb 9, 2017, at 4:46 PM, Melinda Green melinda@superliminal.com [4D_Cu=
bing] <4D_Cubing@yahoogroups.com> wrote:
>=20
> Hello Matt,
>=20
> Great to hear from you again! For the newer members, Matt currently holds=
the record for the shortest 3^4 solution with 227 twists.
>=20
> Good question regarding orientations. No, it hasn't come up yet, and I fe=
el a little embarrassed for not considering it more closely in light of the=
red-blue problem, but then that's what the preview video is for! No, you c=
an't arbitrarily place any piece in any orientation. With this arrangement =
of magnets, each of the red or blue corners can be placed in any of the 3 o=
rientations that leave it in the outer corner position, but no others. I gu=
ess that means the problem is unsolvable with this arrangement of magnets.
>=20
> If I had used the "other" natural arrangement, it would also be possible =
to achieve 3 more orientations by turning a red or blue sticker all the way=
around and sticking it into the center of its 2x2x2 cube. That would be in=
teresting, but would be a completely invalid orientation, leaving a 3-stick=
er junction sticking out of the corner and with the patterns of adjacent pi=
eces not matching at all. By that I mean that the diagonal lines between co=
lors on a piece would meet adjacent pieces in an 'X' configuration rather t=
han matching diagonals. In other words, you could get a mixing of the the n=
ormal view with the crazy inside-out views you see in the video, but I don'=
t think it would be helpful. Those 3 inverted piece orientations are not am=
ong the 12 tetrahedral orientations that you are talking about however. The=
y are from the additional 12 orientations of the cubic pieces, but it is an=
interesting aside.
>=20
> Thanks Matt!
> -Melinda
>=20
>> On 2/9/2017 3:19 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>> Very impressive Melinda!
>>=20
>> Just a quick reply for now, I'll try to find time to think about this in=
more detail. I may have missed this in the video, so apologies if this has=
been addressed already, but have you checked that each piece works in each=
of the 12 possible orientations with the arrangement of magnets? I'm hopin=
g there's some simple way to allow the puzzle to fully scramble.
>>=20
>> Matt
>=20
>=20
--Apple-Mail-FE53156F-3162-4706-8963-014F1892A6F5
Content-Type: text/html;
charset=utf-8
Content-Transfer-Encoding: quoted-printable
=3Dutf-8">
Hi Melinda,
ailSignature">
This is a great achi=
evement!
Signature">I haven't convinced myself this implementation covers all the pe=
rmutations of 2^4. Maybe someone can create a GAP script to quickly verify =
the size of the group as a sanity check.
">
In order to illustrate the idea,=
can you create a 2D version of 2^3 in the same way, using eight square til=
es?
ture">Nan
Sent from my iPhone
"cite">=20
=20=20=20=20=20=20
=20=20=20=20=20=20
=20=20
=20=20
Hello Matt,
Great to hear from you again! For the newer members, Matt currently
holds the record for the shortest 3^4 solution with 227 twists.
Good question regarding orientations. No, it hasn't come up yet, and
I feel a little embarrassed for not considering it more closely in
light of the red-blue problem, but then that's what the preview
video is for! No, you can't arbitrarily place any piece in any
orientation. With this arrangement of magnets, each of the red or
blue corners can be placed in any of the 3 orientations that leave
it in the outer corner position, but no others. I guess that means
the problem is unsolvable with this arrangement of magnets.
If I had used the "other" natural arrangement, it would also be
possible to achieve 3 more orientations by turning a red or blue
sticker all the way around and sticking it into the center of its
2x2x2 cube. That would be interesting, but would be a completely
invalid orientation, leaving a 3-sticker junction sticking out of
the corner and with the patterns of adjacent pieces not matching at
all. By that I mean that the diagonal lines between colors on a
piece would meet adjacent pieces in an 'X' configuration rather than
matching diagonals. In other words, you could get a mixing of the
the normal view with the crazy inside-out views you see in the
video, but I don't think it would be helpful. Those 3 inverted piece
orientations are not among the 12 tetrahedral orientations that you
are talking about however. They are from the additional 12
orientations of the cubic pieces, but it is an interesting aside.
Thanks Matt!
-Melinda
=20=20=20=20=20=20
=20=20=20=20=20=20
Very impressive Melinda!
Just a quick reply for now, I'll try to find time to think about
this in more detail. I may have missed this in the video, so
apologies if this has been addressed already, but have you checked
that each piece works in each of the 12 possible orientations with
the arrangement of magnets? I'm hoping there's some simple way to
allow the puzzle to fully scramble.
Matt
=20=20
=20=20=20=20=20
=20=20=20=20
--Apple-Mail-FE53156F-3162-4706-8963-014F1892A6F5--
From: damienturtle@hotmail.co.uk
Date: Fri, 10 Feb 2017 21:15:50 -0800
Subject: Re: Physical 4D puzzle achieved
--------------DF599BBE03A2326B70D8C80D
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Content-Transfer-Encoding: quoted-printable
Hello Matt,
First off, I'm thrilled that you think the problem is solvable! I like your=
focus on making sure all piece orientations are possible. Your first solut=
ion looks like the right approach. No worries for not understanding the pro=
blem until now. It is one of the important reasons to prototype. Yes, 384 m=
agnets is a lot but it's only double what I'm currently using. It may drive=
the material costs up to $100 but puzzle enthusiasts routinely pay more th=
an that for special puzzles. It does add a scary dimension to the thought o=
f a 3^4 version, but first things first.
Your second solution using free-turning magnetic balls is very clever. I su=
spect it would only be practical if someone already makes such a unit. It d=
oesn't need to be spheres though. It could be ordinary bar magnets on spind=
les. Yet another option occurred to me which is that not all of the embedde=
d parts need to be magnets. Some could be ordinary unmagnetized ferromagnet=
ic material since it will be attracted to both polarities. I don't see how =
to use the idea yet but it's another option.
Regarding moves allowing access to the full state space, I don't fully unde=
rstand your suggestion. It sounds like you are suggesting a restacking move=
where one 2x2x1 block is moved from one end to the other but not flipped, =
is that right? EG starting with the two halves of the gray face visible on =
the ends, all the gray stickers will end up physically touching, just like =
the black face in the center. The problem is that leaves the puzzle in one =
of those strange inverted cases and doesn't seem to move stickers off of th=
e red-blue faces.
One move that it does seem to allow is rotating both end caps 90 degrees on=
to opposite sides of the central 2x2x2. In other words, a 90 degree twist o=
f the central (black) face but in a direction that crucially does move stic=
kers off the red-blue faces. Double moves like this are slightly inelegant,=
but that's a small price to pay for a complete solution. One nice thing ab=
out this twist is that it gives meaning and purpose to the interior physica=
l stickers. Currently, half of the physical stickers are not really needed =
because there are no ordinary twists that leave them exposed. My current ba=
ndaged puzzle would even work better without them.
So thanks for the great ideas. I'm going to see about ordering some pieces =
with your new configuration.
Now here's a mini-puzzle: Can anybody describe a sequence of moves on this =
puzzle that results in reorienting a single piece? Bonus points for short s=
equences that use the fewest double moves. If there's a short enough sequen=
ce, I may show it in the video. One ironic thing is that the fact that you =
can always do it, means that it's valid to simply yank out a single piece a=
nd reorient it directly! Is that cheating or just another shortcut? Interes=
ting stuff.
Happy puzzling!
-Melinda
On 2/10/2017 3:52 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>
>
> Melinda,
>
> It seems that the choice of magnet arrangement isn't correct then. Fortun=
ately, I think that's solvable. I now wish I'd fully understood this approa=
ch when you first discussed it, it took this prototype and some careful thi=
nking before I finally clicked.
>
> First of all, I think that taking a 2x2x1 block from one end and putting =
at the other end is a valid maneuver, a 4D rotation which will allow blue/r=
ed pieces to mix with other colours and to fully scramble the 2^4.
>
> Then, how should the magnets be configured? Well, I see two possible solu=
tions.
>
> 1) We need to be able to pick out any piece and put it back in any of the=
12 orientations of 2^4 corners (which are the rotational symmetries of the=
tetrahedron) and be able to attach it back to the puzzle, which tells us t=
he symmetry of any valid static arrangement of magnets. I skeched a solutio=
n quickly and uploaded it to the group (it's in the group files in my folde=
r since I've no idea how to attach images to a post): shown are 3 sides of =
one piece and 12 magnets embedded in those faces, with the colour of the do=
t indicating the orientation of the magnet. As a quick desciption, there ar=
e 4 magnets on each face in a '+' pattern, with orientation of the magnets =
alternating as you go round the face. This is done such that if the cube is=
rotated about a corner the configuration looks the same. I believe this wi=
ll result in a fully-functioning puzzle, but it takes 384 magnets!
>
> 2) It might be possible to have a spherical magnet in a cavity in the cen=
ter of each of the 6 faces of each piece. When two faces are brought close =
to each other, the magnets should be able to realign so that they attract. =
This only takes 96 magnets, but I'm less convinced it will work well and ro=
bustly than a static configuration of magnets.
>
> If I've made some mistake or someone has a better idea, I'd be interested=
in seeing it. Either way, I hope a physical 2 ^4 is available to buy some =
day, it would be so much fun to play with and show off!
>
> Matt
>
>=20
--------------DF599BBE03A2326B70D8C80D
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">
Hello Matt,
First off, I'm thrilled that you think the problem is solvable! I
like your focus on making sure all piece orientations are possible.
Your first solution looks like the right approach. No worries for
not understanding the problem until now. It is one of the important
reasons to prototype. Yes, 384 magnets is a lot but it's only double
what I'm currently using. It may drive the material costs up to $100
but puzzle enthusiasts routinely pay more than that for special
puzzles. It does add a scary dimension to the thought of a 3^4
version, but first things first.
Your second solution using free-turning magnetic balls is very
clever. I suspect it would only be practical if someone already
makes such a unit. It doesn't need to be spheres though. It could be
ordinary bar magnets on spindles. Yet another option occurred to me
which is that not all of the embedded parts need to be magnets. Some
could be ordinary unmagnetized ferromagnetic material since it will
be attracted to both polarities. I don't see how to use the idea yet
but it's another option.
Regarding moves allowing access to the full state space, I don't
fully understand your suggestion. It sounds like you are suggesting
a restacking move where one 2x2x1 block is moved from one end to the
other but not flipped, is that right? EG starting with the two
halves of the gray face visible on the ends, all the gray stickers
will end up physically touching, just like the black face in the
center. The problem is that leaves the puzzle in one of those
strange inverted cases and doesn't seem to move stickers off of the
red-blue faces.
One move that it does seem to allow is rotating both end caps 90
degrees onto opposite sides of the central 2x2x2. In other words, a
90 degree twist of the central (black) face but in a direction that
crucially does move stickers off the red-blue faces. Double moves
like this are slightly inelegant, but that's a small price to pay
for a complete solution. One nice thing about this twist is that it
gives meaning and purpose to the interior physical stickers.
Currently, half of the physical stickers are not really needed
because there are no ordinary twists that leave them exposed. My
current bandaged puzzle would even work better without them.
So thanks for the great ideas. I'm going to see about ordering some
pieces with your new configuration.
Now here's a mini-puzzle: Can anybody describe a sequence of moves
on this puzzle that results in reorienting a single piece? Bonus
points for short sequences that use the fewest double moves. If
there's a short enough sequence, I may show it in the video. One
ironic thing is that the fact that you can always do it, means that
it's valid to simply yank out a single piece and reorient it
directly! Is that cheating or just another shortcut? Interesting
stuff.
Happy puzzling!
-Melinda
Melinda,
It seems that the choice of magnet arrangement isn't correct then.
Fortunately, I think that's solvable. I now wish I'd fully
understood this approach when you first discussed it, it took this
prototype and some careful thinking before I finally clicked.
First of all, I think that taking a 2x2x1 block from one end and
putting at the other end is a valid maneuver, a 4D rotation which
will allow blue/red pieces to mix with other colours and to fully
scramble the 2^4.
Then, how should the magnets be configured? Well, I see two
possible solutions.
1) We need to be able to pick out any piece and put it back in any
of the 12 orientations of 2^4 corners (which are the rotational
symmetries of the tetrahedron) and be able to attach it back to
the puzzle, which tells us the symmetry of any valid static
arrangement of magnets. I skeched a solution quickly and uploaded
it to the group (it's in the group files in my folder since I've
no idea how to attach images to a post): shown are 3 sides of one
piece and 12 magnets embedded in those faces, with the colour of
the dot indicating the orientation of the magnet. As a quick
desciption, there are 4 magnets on each face in a '+' pattern,
with orientation of the magnets alternating as you go round the
face. This is done such that if the cube is rotated about a corner
the configuration looks the same. I believe this will result in a
fully-functioning puzzle, but it takes 384 magnets!
2) It might be possible to have a spherical magnet in a cavity in
the center of each of the 6 faces of each piece. When two faces
are brought close to each other, the magnets should be able to
realign so that they attract. This only takes 96 magnets, but I'm
less convinced it will work well and robustly than a static
configuration of magnets.
If I've made some mistake or someone has a better idea, I'd be
interested in seeing it. Either way, I hope a physical 2 ^4 is
available to buy some day, it would be so much fun to play with
and show off!
Matt
=20=20=20=20=20=20
--------------DF599BBE03A2326B70D8C80D--
From: Christopher Locke <project.eutopia@gmail.com>
Date: Sat, 11 Feb 2017 01:53:08 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--------------1B75BCF845CC96A491186702
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I figured out some more possible twists. See my attached images.
The first is a labelling of the 2x2x2x2 cube in MagicCube4D. The -w
face is central, surrounded by the +/- x, y, z faces with +w hidden.
The second shows the axis definitions of the physical cube. The
leftmost 8 cubies those part of the -w face, and the rightmost part of
the +w face.
If we label a simple 90 degree twist by first specifying the face, then
the axis to rotate about (e.g. -w:+z rotates the central -w block around
the +z axis), then we can investigate some twists.
- Any twist of the form +/-w:+/-(x,y,z) is done on the physical puzzle
by rotating the +/-w block (left or right 2x2x2 cube) about the
respective x,y,z axis. This is the easiest twist to make. Essentially
you can detach and freely rotate the left or right blocks any way you
desire, and it is a valid twist.
- Any twist of the form +/-(x,y,z):+/-w can also be done relatively
easily. For instance, the +z:+w twist involves doing U' on left block
and U on right block. +y:+w is R on left block and L' on right block.
All possibilities involve twisting one block with some normal Rubik's
cube rotation, and the mirrored twist on the other block.
- Putting these two moves together, and you are allowed to do any
rotation of the left and right blocks, or do a single 90 degree regular
Rubik's cube twist to both left and right simultaneously (do not need to
do the exact mirror, because w:(x,y,z) twists can put left/right blocks
in any orientation). So from this point of view, it is like solving two
2^3 cubes by doing any even number of 90 degree twists to each block, or
an odd number to each block (can always do then undo a move on one side,
so it is the parity of the number of twists that must remain
unchanged). This is the state of the puzzle without being able to mix
up red/blue (+/-w face) stickers with the other stickers.
To make the puzzle more 4D, you need to be able to mix up w and (x,y,z)
stickers, and this can only be done using twists that involve only x,y,z
axes. However, if you can figure out how one such twist works, then
reorientations of the physical puzzle (rotate the left/right physical
block any way, while doing the mirror rotation to the other block) can
put it into an orientation to get any twist you desire. Let's focus on
the +x:+z twist.
This twist moves the frontmost 8 pieces of the physical puzzle to the
left, cyclically (so those on the left are brought all the way to the
right). One 90 degree +x:+z twist of the 4D puzzle involves moving
these physical pieces to the left, while also rotating each cubie 180
degrees. The 180 degree twist is about a diagonal line in the "x-z"
line, as drawn roughly at the bottom of the 2nd figure. The bottom
figure shows the rotation for one of the cubies in the top half. The
cubies in the bottom have the same 180 degree rotation, but mirrored
along the x-y plane that splits top/bottom. This mixes up w-face and
y-face stickers.
All such "difficult" twists I think are accomplished by moving the
respective 8 physical cubies around cyclically, while also doing some
180 degree twist of the individual cubies. I imagine these would be
fairly annoying to do on the physical cube, so it would be worth
investigating if some of those "inversion" moves Melinda showed can be
combined with simpler moves in a commutator/conjugation to give an
equivalent x,y,z twist.
Best regards,
Chris
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--------------1B75BCF845CC96A491186702--
From: Christopher Locke <project.eutopia@gmail.com>
Date: Sat, 11 Feb 2017 19:00:13 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
To continue a bit of my previous line of thought. I considered how this
would work in 3D case: http://imgur.com/a/sUqEh
Essentially my conclusion is that in the 3D case being reduced to two 2D
squares, there is no rigid orientation of stickers that allow U, D, and
R moves to work using clean rotations of the physical 2D squares. You
can get it to allow 90 degree twists of U, D, and 180 degree twists of
F, B, R, L simply, but there is some necessary "cubie twisting" that is
required to implement F, B, R, L twists (e.g. to do a R twist, you need
to rotate the middle 4 square cubies around by 90 degrees, while also
twisting each cubie to mix up x- and z-face stickers). This is
analogous to the 4D case Melinda created, where 90 degree twists that
mix up red/blue (w-face stickers) with other stickers requires
simultaneous twisting of the cubies themselves too.
It might be possible if the stickers themselves were slightly
free-floating within the cubie, and somehow the internal orientation of
the stickers could be moved during twists. I illustrated this idea with
the 2D representation of the 3D case in the second image. I analyzed the
R twist, by breaking it down into the rigid 90 degree rotation, and the
necessary internal twist of stickers. If you split each square cubie
into 120 degree sticker "sectors", then when you do the R rotation, you
need to rotate the internal stickers also by +/- 75 degrees to keep the
stickers in the correct location.
These considerations imply to me that it is impossible to make a simple
2D representation of the 3D puzzle as two squares that has the same
simple 90 degree rotations possible as the original 3D puzzle. This can
only be done by having some very clever stickers that are "floating"
within each cubie and that rotate correctly to keep stickers in the
correct position during 90 degree twists. This could possibly be done
by having slightly rounded stickers that can roll over each other in
some nice way to allow both U or D twists, and the central R twist.
Note that since the internal twists are alternating between 75 and -75
degrees, a 180 degree twist is okay still, just like on the physical 4D
puzzle.
Best regards,
Chris
From: damienturtle@hotmail.co.uk
Date: 12 Feb 2017 20:44:15 +0000
Subject: Re: Physical 4D puzzle achieved
From: damienturtle@hotmail.co.uk
Date: Sun, 12 Feb 2017 14:53:43 -0800
Subject: Re: Physical 4D puzzle achieved
--------------73EA5800E3C834A99706116F
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In my last message I mentioned that it appears that moving both end caps to opposite sides (as opposed to "ends") works. That looks to be a 90 degree rotation of the central black face, and exchanges red-blue stickers with other colors. Can someone confirm or refute that? What surprises me is just how scrambled the puzzle looks after just one of these moves. It clearly makes the puzzle much more difficult. It also suggests that to solve it, you'll probably need to frequently examine the interior. If so, perhaps the ultimate version would involve transparent materials such as colored glass, or perhaps clear cubes with four colored corners. It also makes the current bandaged version look not so bad. I suspect a natural solution would first position and orient all the red and blue stickers and then solve the bandaged form.
-Melinda
On 2/12/2017 12:44 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>
>
> I've looked at this a little more today, and have a correction to make to my previous message.
>
> >First of all, I think that taking a 2x2x1 block from one end and putting at the other end is a valid maneuver, a 4D rotation which will allow blue/red pieces to mix with other colours and to fully scramble the 2^4.
>
> Nope, sadly it is not quite that simple. I'm now convinced that there is no simple maneuver which will do a 4D rotation to put some other pair of colours in place of blue/red, or a twist which mixes red/blue stickers with other colours. There are of course ways to achieve these, but they require several steps and I'm not yet sure what the easiest/shortest one would be, or how best to describe any such maneuver. I'll post an update if I find a good answer, unless someone else figures it out first.
>
> Matt
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In my last message I mentioned that it appears that moving both end
caps to opposite sides (as opposed to "ends") works. That looks to
be a 90 degree rotation of the central black face, and exchanges
red-blue stickers with other colors. Can someone confirm or refute
that? What surprises me is just how scrambled the puzzle looks after
just one of these moves. It clearly makes the puzzle much more
difficult. It also suggests that to solve it, you'll probably need
to frequently examine the interior. If so, perhaps the ultimate
version would involve transparent materials such as colored glass,
or perhaps clear cubes with four colored corners. It also makes the
current bandaged version look not so bad. I suspect a natural
solution would first position and orient all the red and blue
stickers and then solve the bandaged form.
-Melinda
I've looked at this a little more today, and have a correction to
make to my previous message.
>First of all, I think that taking a 2x2x1 block from one end
and putting at the other end is a valid maneuver, a 4D rotation
which will allow blue/red pieces to mix with other colours and to
fully scramble the 2^4.
Nope, sadly it is not quite that simple. I'm now convinced that
there is no simple maneuver which will do a 4D rotation to put
some other pair of colours in place of blue/red, or a twist which
mixes red/blue stickers with other colours. There are of course
ways to achieve these, but they require several steps and I'm not
yet sure what the easiest/shortest one would be, or how best to
describe any such maneuver. I'll post an update if I find a good
answer, unless someone else figures it out first.
Matt
--------------73EA5800E3C834A99706116F--
From: damienturtle@hotmail.co.uk
Date: 12 Feb 2017 23:46:17 +0000
Subject: Re: Physical 4D puzzle achieved
From: damienturtle@hotmail.co.uk
Date: Sun, 12 Feb 2017 15:35:32 -0800
Subject: Re: Physical 4D puzzle achieved
--------------01F2DCEF128B7295AD844F4D
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Matt,
>First of all, I think that taking a 2x2x1 block from one end and
putting at the other end is a valid maneuver, a 4D rotation which will
allow blue/red pieces to mix with other colours and to fully scramble
the 2^4.
It is possible to swap two 2x2x1 blocks, if you are careful to keep the
orientations correct. For instance swapping the two topmost 2x2x1
blocks (as demonstrated in Melinda's video) is a 180 degree rotation of
the +z face about the y axis (it moves cubies of +z face, keeps +z
stickers on +z face, +y stickers on +y face, and swaps the +/-w
(red/blue) sticker positions and the +/-x positions). Similarly,
swapping the front face of the physical puzzle is a 180 degree rotation
of the +x face about the y axis.
However, red/blue (w-face) stickers are always pointing "outwards",
while the x, y, z colors are pointing towards the 2x2x2 cube-half faces,
so these moves do not mix up red/blue with the others.
Melinda,
Could you elaborate on what kind of move you are describing? I cannot
visualize it very well.
Also, I don't think there is a need to make the puzzle transparent, as
all 4 stickers attached to each cubie are always visible (one on the
outward facing corner, and three touching 2x2x2 cube face centers).
By the way, looking at the video again, the move at 23:30
(https://youtu.be/Asx653BGDWA?t=1410) could be a valid remixing of the
red/blue. It is hard to check by hand by just looking at the video
though, as I cannot easily enumerate how each piece changed. It is
definitely not a single 90 degree twist, but it could be a combination
of twists which would open the possibility of having all permutations
available. My rough feeling by looking at it is that it is doing a "90
degree" twist of the top and bottom, "90 degree" twist of the front and
back, and "90 degree" twist through the inside/outside faces and the
red/blue w-faces. It could of course be an illegal position due to
parity or something, but that would require deeper investigation.
Probably the easiest way would be to put the positions into MagicCube4D
and then try to solve it from that configuration :D
Best regards,
Chris
On 2017年02月12日 14:53, Melinda Green melinda@superliminal.com [4D_Cubing]
wrote:
>
> In my last message I mentioned that it appears that moving both end
> caps to opposite sides (as opposed to "ends") works. That looks to be
> a 90 degree rotation of the central black face, and exchanges red-blue
> stickers with other colors. Can someone confirm or refute that? What
> surprises me is just how scrambled the puzzle looks after just one of
> these moves. It clearly makes the puzzle much more difficult. It also
> suggests that to solve it, you'll probably need to frequently examine
> the interior. If so, perhaps the ultimate version would involve
> transparent materials such as colored glass, or perhaps clear cubes
> with four colored corners. It also makes the current bandaged version
> look not so bad. I suspect a natural solution would first position and
> orient all the red and blue stickers and then solve the bandaged form.
>
> -Melinda
>
> On 2/12/2017 12:44 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>> I've looked at this a little more today, and have a correction to
>> make to my previous message.
>>
>> >First of all, I think that taking a 2x2x1 block from one end and
>> putting at the other end is a valid maneuver, a 4D rotation which
>> will allow blue/red pieces to mix with other colours and to fully
>> scramble the 2^4.
>>
>> Nope, sadly it is not quite that simple. I'm now convinced that there
>> is no simple maneuver which will do a 4D rotation to put some other
>> pair of colours in place of blue/red, or a twist which mixes red/blue
>> stickers with other colours. There are of course ways to achieve
>> these, but they require several steps and I'm not yet sure what the
>> easiest/shortest one would be, or how best to describe any such
>> maneuver. I'll post an update if I find a good answer, unless someone
>> else figures it out first.
>>
>> Matt
>
>
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Matt,
>First of all, I think that taking a 2x2x1 block from one end
and putting at the other end is a valid maneuver, a 4D rotation
which will allow blue/red pieces to mix with other colours and to
fully scramble the 2^4.
It is possible to swap two 2x2x1 blocks, if you are careful to
keep the orientations correct. For instance swapping the two
topmost 2x2x1 blocks (as demonstrated in Melinda's video) is a 180
degree rotation of the +z face about the y axis (it moves cubies
of +z face, keeps +z stickers on +z face, +y stickers on +y face,
and swaps the +/-w (red/blue) sticker positions and the +/-x
positions). Similarly, swapping the front face of the physical
puzzle is a 180 degree rotation of the +x face about the y axis.
However, red/blue (w-face) stickers are always pointing
"outwards", while the x, y, z colors are pointing towards the
2x2x2 cube-half faces, so these moves do not mix up red/blue with
the others.
Melinda,
Could you elaborate on what kind of move you are describing? I
cannot visualize it very well.
Also, I don't think there is a need to make the puzzle
transparent, as all 4 stickers attached to each cubie are always
visible (one on the outward facing corner, and three touching
2x2x2 cube face centers).
By the way, looking at the video again, the move at 23:30
(https://youtu.be/Asx653BGDWA?t=1410) could be a valid remixing of
the red/blue. It is hard to check by hand by just looking at the
video though, as I cannot easily enumerate how each piece
changed. It is definitely not a single 90 degree twist, but it
could be a combination of twists which would open the possibility
of having all permutations available. My rough feeling by looking
at it is that it is doing a "90 degree" twist of the top and
bottom, "90 degree" twist of the front and back, and "90 degree"
twist through the inside/outside faces and the red/blue w-faces.Â
It could of course be an illegal position due to parity or
something, but that would require deeper investigation. Probably
the easiest way would be to put the positions into MagicCube4D and
then try to solve it from that configuration :D
Best regards,
Chris
cite="mid:56e0f291-fbdf-3943-8295-ee618e40ccc7@superliminal.com"
type="cite">
 In my last message I mentioned that it appears that
moving both end caps to opposite sides (as opposed to
"ends") works. That looks to be a 90 degree rotation of
the central black face, and exchanges red-blue stickers
with other colors. Can someone confirm or refute that?
What surprises me is just how scrambled the puzzle looks
after just one of these moves. It clearly makes the puzzle
much more difficult. It also suggests that to solve it,
you'll probably need to frequently examine the interior.
If so, perhaps the ultimate version would involve
transparent materials such as colored glass, or perhaps
clear cubes with four colored corners. It also makes the
current bandaged version look not so bad. I suspect a
natural solution would first position and orient all the
red and blue stickers and then solve the bandaged form.
-Melinda
On 2/12/2017 12:44 PM,
moz-do-not-send="true" class="moz-txt-link-abbreviated"
href="mailto:damienturtle@hotmail.co.uk">damienturtle@hotmail.co.uk
[4D_Cubing] wrote:
type="cite"> I've looked at this a little more today, and
have a correction to make to my previous message.
>First of all, I think that taking a 2x2x1 block from
one end and putting at the other end is a valid maneuver,
a 4D rotation which will allow blue/red pieces to mix with
other colours and to fully scramble the 2^4.
Nope, sadly it is not quite that simple. I'm now convinced
that there is no simple maneuver which will do a 4D
rotation to put some other pair of colours in place of
blue/red, or a twist which mixes red/blue stickers with
other colours. There are of course ways to achieve these,
but they require several steps and I'm not yet sure what
the easiest/shortest one would be, or how best to describe
any such maneuver. I'll post an update if I find a good
answer, unless someone else figures it out first.
Matt
--------------01F2DCEF128B7295AD844F4D--
From: Melinda Green <melinda@superliminal.com>
Date: Sun, 12 Feb 2017 17:55:42 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--------------14CB8241AA2547E65FF181CE
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Matt,
Christopher's messages were a bit to opaque to me but I'm starting to get t=
he gist. It's good to know that this 90 degree twist is a valid move though=
it's unfortunate that it's far from pure. These almost look like gear-cube=
twists now. I even think I can guess how the orientations are supposed to =
end up after the appropriate reorientations of the black pieces. (Alternati=
ng CW and CCW twists of each piece about their black stickers.) I've attach=
ed a sequence of snaps showing the process. (Also here .com/misc/twist90cp.jpg> in case the attachment doesn't work.) The second s=
nap shows the twist in progress. The third shows it completed, with me hold=
ing it in place against the magnets. You can see what I mean about the puzz=
le looking completely scrambled by this one twist. The fourth snap shows it=
with all 8 of the twisted pieces reoriented. The interesting thing is how =
it results in a much less scrambled looking puzzle.
Christopher,
I hope the photos helped. One interesting to note is that the end result of=
the sequence (plus a simple rotation) resembles the result of the double s=
wap you highlighted in the video (https://youtu.be/Asx653BGDWA?t=3D1410) so=
maybe there's hope for a more practical way to reach the full 2^4 state sp=
ace.
Thanks!
-Melinda
On 2/12/2017 3:46 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
> Melinda,
>
> > In my last message I mentioned that it appears that moving both end cap=
s to opposite sides (as opposed to "ends") works. That looks to be a 90 deg=
ree rotation of the central black face, and exchanges red-blue stickers wit=
h other colors. Can someone confirm or refute that?
>
> I realised my previous post might not be clear on a few things, so I want=
to address this explicitly. If I understand correctly (please correct me o=
therwise), you are taking the two 2x2x1 end caps off, rotating (not twistin=
g) the 'black cube' and then replacing the end caps, such that the red and =
blue stickers mix with other stickers. This rotation of the black face on t=
he physical puzzle seems tempting and I thought it was a valid, normal twis=
t of the 4D puzzle at first glance, but I now realise I was wrong.
>
> However, it does lead to a legal configuration, and corresponds to perfor=
ming a normal twist of the black face then rotating the 8 corners of that f=
ace in a certain way. It can be performed in 9 twists in MC4D, though only =
5 if 180 twists could be done in one click, and is equivalent to the 2x2x2 =
position obtained by performing U2 R2 F2 U2 on a solved cube.
>
> It might not correspond to a normal twist on the 2^4, but it seems like i=
t would lead to an interesting solving experience in its own right.
>
> Also, if the black face is rotated such that the red and blue stickers do=
n't mix with other stickers, which is the same as rotating the middle 2x2x2=
about the long axis, then it is a valid twist, no problems.
>
> I've only skimmed through Christopher's posts on the subject, but I shoul=
d mention that he seems to have come to the same conclusion independently, =
which is reassuring.
>
> Matt
>
>=20
--------------FF6313DC3272E5CB35962A3D
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable
">
Matt,
Christopher's messages were a bit to opaque to me but I'm starting
to get the gist. It's good to know that this 90 degree twist is a
valid move though it's unfortunate that it's far from pure. These
almost look like gear-cube twists now. I even think I can guess how
the orientations are supposed to end up after the appropriate
reorientations of the black pieces. (Alternating CW and CCW twists
of each piece about their black stickers.) I've attached a sequence
of snaps showing the process. ( href=3D"http://superliminal.com/misc/twist90cp.jpg">Also here in
case the attachment doesn't work.) The second snap shows the twist
in progress. The third shows it completed, with me holding it in
place against the magnets. You can see what I mean about the puzzle
looking completely scrambled by this one twist. The fourth snap
shows it with all 8 of the twisted pieces reoriented. The
interesting thing is how it results in a much less scrambled looking
puzzle.
Christopher,
I hope the photos helped. One interesting to note is that the end
result of the sequence (plus a simple rotation) resembles the result
of the double swap you highlighted in the video ( class=3D"moz-txt-link-freetext"
href=3D"https://youtu.be/Asx653BGDWA?t=3D1410">https://youtu.be/Asx65=
3BGDWA?t=3D1410)
so maybe there's hope for a more practical way to reach the full 2^4
state space.
Thanks!
-Melinda
On 2/12/2017 3:46 PM, lto:damienturtle@hotmail.co.uk">damienturtle@hotmail.co.uk [4D_Cubing] =
wrote:
--------------FF6313DC3272E5CB35962A3D--
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Content-Transfer-Encoding: 7bit
Content-Type: image/jpeg;
name="twist90cp.jpg"
Content-Transfer-Encoding: base64
Content-Disposition: attachment;
filename="twist90cp.jpg"
--------------14CB8241AA2547E65FF181CE--
From: Melinda Green <melinda@superliminal.com>
Date: Sun, 12 Feb 2017 19:35:01 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--------------1A92EE1249B9853DE1FDDF08
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: quoted-printable
Yes, the reorientations were all 120 degree twists about the long diagonals=
through their black stickers. I knew the black stickers had to remain wher=
e they were because it was a twist of the black face.
The magnets do not want to allow the twist or the piece reorientations, tho=
ugh the central 2x2x2 cube was happy once it was fully reorientated. The fr=
ames where you see my hand are the configurations that the magnets do not a=
llow. Matt's pattern should allow everything.
-Melinda
On 2/12/2017 7:23 PM, Christopher Locke project.eutopia@gmail.com [4D_Cubin=
g] wrote:
>
>
> Melinda,
>
> Yes, that looks like a correct twist of the +y hyperface about the +z axi=
s! If the colors are labelled: (-x brown, +x purple, -y gray, +y black, -z=
light blue, +z green, -w blue, +w red), then that 90 degree twist should m=
ove those middle 8 physical cubies around in a 90 degree twist just like yo=
u did, and the x/w stickers should move blue -> purple -> red -> brown. Fr=
om step 3 to 4, I take it you did a 120 degree twist about a diagonal axis =
that goes through the center of each cubie and the total center of the phys=
ical puzzle (where black stickers are)?
>
> By the way, was the magnetic orientation okay after doing those twists? =
In the video move I pointed out (https://youtu.be/Asx653BGDWA?t=3D1410), yo=
u had problems doing some twists after the double inversion due to magnet p=
ositioning.
>
> Best regards,
> Chris
>
> On 2017=E5=B9=B402=E6=9C=8812=E6=97=A5 17:55, Melinda Green melinda@super=
liminal.com [4D_Cubing] wrote:
>>
>> Matt,
>>
>> Christopher's messages were a bit to opaque to me but I'm starting to ge=
t the gist. It's good to know that this 90 degree twist is a valid move tho=
ugh it's unfortunate that it's far from pure. These almost look like gear-c=
ube twists now. I even think I can guess how the orientations are supposed =
to end up after the appropriate reorientations of the black pieces. (Altern=
ating CW and CCW twists of each piece about their black stickers.) I've att=
ached a sequence of snaps showing the process. (Also here nal.com/misc/twist90cp.jpg> in case the attachment doesn't work.) The secon=
d snap shows the twist in progress. The third shows it completed, with me h=
olding it in place against the magnets. You can see what I mean about the p=
uzzle looking completely scrambled by this one twist. The fourth snap shows=
it with all 8 of the twisted pieces reoriented. The interesting thing is h=
ow it results in a much less scrambled looking puzzle.
>>
>> Christopher,
>>
>> I hope the photos helped. One interesting to note is that the end result=
of the sequence (plus a simple rotation) resembles the result of the doubl=
e swap you highlighted in the video (https://youtu.be/Asx653BGDWA?t=3D1410)=
so maybe there's hope for a more practical way to reach the full 2^4 state=
space.
>>
>> Thanks!
>> -Melinda
>>
>> On 2/12/2017 3:46 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>>
>>
>
>
>
>=20
--------------1A92EE1249B9853DE1FDDF08
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable
">
Yes, the reorientations were all 120 degree twists about the long
diagonals through their black stickers. I knew the black stickers
had to remain where they were because it was a twist of the black
face.
The magnets do not want to allow the twist or the piece
reorientations, though the central 2x2x2 cube was happy once it was
fully reorientated. The frames where you see my hand are the
configurations that the magnets do not allow. Matt's pattern should
allow everything.
-Melinda
cite=3D"mid:9fe43e0f-d1e5-43ba-d42f-99d5ba6a5084@gmail.com"
type=3D"cite">
pe">
=20=20=20=20=20=20
Melinda,
Yes, that looks like a correct twist of the +y hyperface about
the +z axis!=C2=A0 If the colors are labelled: (-x brown, +x purple=
,
-y gray, +y black, -z light blue, +z green, -w blue, +w red),
then that 90 degree twist should move those middle 8 physical
cubies around in a 90 degree twist just like you did, and the
x/w stickers should move blue -> purple -> red ->
brown.=C2=A0 From step 3 to 4, I take it you did a 120 degree twist
about a diagonal axis that goes through the center of each cubie
and the total center of the physical puzzle (where black
stickers are)?
By the way, was the magnetic orientation okay after doing those
twists?=C2=A0 In the video move I pointed out ( moz-do-not-send=3D"true" class=3D"moz-txt-link-freetext"
href=3D"https://youtu.be/Asx653BGDWA?t=3D1410">https://youtu.be/A=
sx653BGDWA?t=3D1410),
you had problems doing some twists after the double inversion
due to magnet positioning.
Best regards,
Chris
On 2017=E5=B9=B402=E6=9C=8812=E6=97=A5=
17:55, Melinda Green
moz-do-not-send=3D"true" class=3D"moz-txt-link-abbreviated"
href=3D"mailto:melinda@superliminal.com">melinda@superliminal.com=
[4D_Cubing] wrote:
cite=3D"mid:bc49bbed-9629-3a99-5a46-06900a36a387@superliminal.com"
type=3D"cite"> =C2=A0
Matt,
Christopher's messages were a bit to opaque to me but I'm
starting to get the gist. It's good to know that this 90
degree twist is a valid move though it's unfortunate that
it's far from pure. These almost look like gear-cube twists
now. I even think I can guess how the orientations are
supposed to end up after the appropriate reorientations of
the black pieces. (Alternating CW and CCW twists of each
piece about their black stickers.) I've attached a sequence
of snaps showing the process. ( href=3D"http://superliminal.com/misc/twist90cp.jpg">Also
here in case the attachment doesn't work.) The second
snap shows the twist in progress. The third shows it
completed, with me holding it in place against the magnets.
You can see what I mean about the puzzle looking completely
scrambled by this one twist. The fourth snap shows it with
all 8 of the twisted pieces reoriented. The interesting
thing is how it results in a much less scrambled looking
puzzle.
Christopher,
I hope the photos helped. One interesting to note is that
the end result of the sequence (plus a simple rotation)
resembles the result of the double swap you highlighted in
the video ( class=3D"moz-txt-link-freetext"
href=3D"https://youtu.be/Asx653BGDWA?t=3D1410">https://youtu.=
be/Asx653BGDWA?t=3D1410)
so maybe there's hope for a more practical way to reach the
full 2^4 state space.
Thanks!
-Melinda
On 2/12/2017 3:46 PM, class=3D"moz-txt-link-abbreviated"
href=3D"mailto:damienturtle@hotmail.co.uk">damienturtle@hotma=
il.co.uk
[4D_Cubing] wrote:
type=3D"cite">
=20=20=20=20=20=20
--------------1A92EE1249B9853DE1FDDF08--
From: Roice Nelson <roice3@gmail.com>
Date: Wed, 15 Feb 2017 11:20:06 -0600
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--f40304361cee623804054894e435
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: quoted-printable
What an awesome puzzle and flurry of activity! This is really cool
Melinda, and there is so much to think about. I especially find the
limitations imposed by the magnets intriguing. I've been out of town and
missed much of the real-time fun, but read through the thread and watched
Mathologer's video last night. I must confess I have not fully digested
all the great ideas flying around, but I had a few thoughts that still seem
useful to share.
Like Nan, my first instinct was to understand the lower dimensional case,
and I'm glad I tried. I started to think of a 4x2 block of squares, and it
took me a bit to realize that was making it more difficult to think about.
It may be obvious to folks, but my Aha moment was that each piece of
Melinda's puzzle is really a face of the dual polytope, the 16-cell
. In the lower dimensional case,
each piece is a face of the octahedron. So it is natural to make the
2^3 analogue a set of triangles, a "net
"
of them in fact. This all made me realize Melinda's 16 pieces are
representing tetrahedral faces, again a net
of the
16-cell.
I think the brilliance of Melinda's design is that each tetrahedron is
represented by a cube, so the net becomes this nice 4x2x2 block. A less
elegant way to approach this puzzle would be to use the net of 16
tetrahedra directly, which should work even if more awkward. Using cubes
for tetrahedra is possible because the tetrahedral group is a subgroup of
the octahedral group. This doesn't work in the lower dimensional case
because a triangle group is not a subgroup of a square group (look up cycli=
c
groups if you want to
research). That is why it was unnatural to deal with squares for the 2^3
analogue.
So now I'm considering Melinda's puzzle as a net of the 16-cell, which
makes it easier to think about what general rotations and twists are.
- A rotation is any detachment of a connected subset of cells, and
subsequent reattachment that preserves the structure of the net.
- A twist is any planar cut of the net into two equal parts, followed by
an arbitrary reorientation of one of the halves (and optionally one coul=
d
also make net preserving changes to the half too), then a reattachment b=
ack
along the planar cut.
Given that, I thought I'd highlight a couple twists I don't think I've seen
yet:
1. A slight modification of Melinda's reorienting move... Pull apart
the two halves, but leave one fixed and rotate the second 90 degrees and
reattach.
2. If we were to allow interim jumbling, I think we can get 90-degree
twists of the blue and orange faces. Instead of performing a
180-degree rotation maneuver here, you would take the 4x2 block and
translate it a step. The end result would be a 3x2x2 with two 2x1 block=
s
protruding off of it on opposite sides. But then you could just do a
reorientation by rolling one of those protrusions around to meet the oth=
er
and recover the original 4x2x2 block. Hope this is clear.
I want one Melinda! When are you going to set up a shop? :D
Cheers,
Roice
On Sun, Feb 12, 2017 at 9:35 PM, Melinda Green melinda@superliminal.com
[4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:
>
>
> Yes, the reorientations were all 120 degree twists about the long
> diagonals through their black stickers. I knew the black stickers had to
> remain where they were because it was a twist of the black face.
>
> The magnets do not want to allow the twist or the piece reorientations,
> though the central 2x2x2 cube was happy once it was fully reorientated. T=
he
> frames where you see my hand are the configurations that the magnets do n=
ot
> allow. Matt's pattern should allow everything.
>
> -Melinda
>
> On 2/12/2017 7:23 PM, Christopher Locke project.eutopia@gmail.com
> [4D_Cubing] wrote:
>
> Melinda,
>
> Yes, that looks like a correct twist of the +y hyperface about the +z
> axis! If the colors are labelled: (-x brown, +x purple, -y gray, +y blac=
k,
> -z light blue, +z green, -w blue, +w red), then that 90 degree twist shou=
ld
> move those middle 8 physical cubies around in a 90 degree twist just like
> you did, and the x/w stickers should move blue -> purple -> red -> brown.
> From step 3 to 4, I take it you did a 120 degree twist about a diagonal
> axis that goes through the center of each cubie and the total center of t=
he
> physical puzzle (where black stickers are)?
>
> By the way, was the magnetic orientation okay after doing those twists?
> In the video move I pointed out (https://youtu.be/Asx653BGDWA?t=3D1410),
> you had problems doing some twists after the double inversion due to magn=
et
> positioning.
>
> Best regards,
> Chris
> On 2017=E5=B9=B402=E6=9C=8812=E6=97=A5 17:55, Melinda Green melinda@super=
liminal.com [4D_Cubing]
> wrote:
>
>
>
> Matt,
>
> Christopher's messages were a bit to opaque to me but I'm starting to get
> the gist. It's good to know that this 90 degree twist is a valid move
> though it's unfortunate that it's far from pure. These almost look like
> gear-cube twists now. I even think I can guess how the orientations are
> supposed to end up after the appropriate reorientations of the black
> pieces. (Alternating CW and CCW twists of each piece about their black
> stickers.) I've attached a sequence of snaps showing the process. (Also
> here in case the attachment
> doesn't work.) The second snap shows the twist in progress. The third sho=
ws
> it completed, with me holding it in place against the magnets. You can se=
e
> what I mean about the puzzle looking completely scrambled by this one
> twist. The fourth snap shows it with all 8 of the twisted pieces
> reoriented. The interesting thing is how it results in a much less
> scrambled looking puzzle.
>
> Christopher,
>
> I hope the photos helped. One interesting to note is that the end result
> of the sequence (plus a simple rotation) resembles the result of the doub=
le
> swap you highlighted in the video (https://youtu.be/Asx653BGDWA?t=3D1410)
> so maybe there's hope for a more practical way to reach the full 2^4 stat=
e
> space.
>
> Thanks!
> -Melinda
>
> On 2/12/2017 3:46 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>
>
>
>
>
>
>=20
>
--f40304361cee623804054894e435
Content-Type: text/html; charset=UTF-8
Content-Transfer-Encoding: quoted-printable
What an awesome puzzle and flurry of activity!=C2=A0 This =
is really cool Melinda, and there is so much to think about.=C2=A0 I especi=
ally find the limitations imposed by the magnets intriguing.=C2=A0 I've=
been out of town and missed much of the real-time fun, but read through th=
e thread and watched Mathologer's video last night.=C2=A0 I must confes=
s I have not fully digested all the great ideas flying around, but I had a =
few thoughts that still seem useful to share.
Like Nan, =
my first instinct was to understand the lower dimensional case, and I'm=
glad I tried.=C2=A0 I started to think of a 4x2 block of squares, and it t=
ook me a bit to realize that was making it more difficult to think about.=
=C2=A0 It may be obvious to folks, but my Aha moment was that each piece of=
Melinda's puzzle is really a face of the dual polytope, the=C2=A0
ef=3D"https://en.wikipedia.org/wiki/16-cell" target=3D"_blank">16-cell.=
=C2=A0 In the lower dimensional case, each piece is a face of the octahedro=
n.=C2=A0 So it is natural to make the 2^3=C2=A0analogue=C2=A0a set of trian=
gles, a "
le:Octahedron_flat.svg" target=3D"_blank">net" of them in fact.=C2=
=A0 This all made me realize Melinda's 16 pieces are representing tetra=
hedral faces, again a
ia/File:16-cell_net.png" target=3D"_blank">net=C2=A0of the 16-cell. =C2=
=A0
I think the brilliance of Melinda's design=
is that each tetrahedron is represented by a cube, so the net becomes this=
nice 4x2x2 block.=C2=A0 A less elegant way to approach this puzzle would b=
e to use the net of 16 tetrahedra directly, which should work even if more =
awkward.=C2=A0 Using cubes for tetrahedra is possible because the tetrahedr=
al group is a subgroup of the octahedral group.=C2=A0 This doesn't work=
in the lower dimensional case because a triangle group is not a subgroup o=
f a square group (look up=C2=A0
lic_group" target=3D"_blank">cyclic groups if you want to research).=C2=
=A0 That is why it was unnatural to deal with squares for the 2^3 analogue.=
So now I'm considering Melinda's puzzle a=
s a net of the 16-cell, which makes it easier to think about what general r=
otations and twists are. =C2=A0
- A rotation is any detachm=
ent of a connected subset of cells, and subsequent reattachment that preser=
ves the structure of the net.=C2=A0=C2=A0 - A twist is any planar cut=
of the net into two equal parts, followed by an arbitrary reorientation of=
one of the halves (and optionally one could also make net preserving chang=
es to the half too), then a reattachment back along the planar cut. l>
Given that, I thought I'd highlight a couple twists I don&=
#39;t think I've seen yet:
- A slight modification of M=
elinda's reorienting move...=C2=A0 Pull apart the two halves, but leave=
one fixed and rotate the second 90 degrees and reattach.
- If we=
were to allow interim jumbling, I think we can get 90-degree twists of the=
blue and orange faces.=C2=A0 Instead of performing=C2=A0a 180-degree=C2=A0=
rotation maneuver=C2=A0here, you would take the 4x2 block and translate it =
a step.=C2=A0 The end result would be a 3x2x2 with two 2x1 blocks protrudin=
g off of it on opposite sides.=C2=A0 But then you could just do a reorienta=
tion by rolling one of those protrusions around to meet the other and recov=
er the original 4x2x2 block.=C2=A0 Hope this is clear.
=
I want one Melinda! When are you going to set up a shop? :D
<=
/div>
Cheers,
Roice
<=
br>
--f40304361cee623804054894e435--
From: Melinda Green <melinda@superliminal.com>
Date: Wed, 15 Feb 2017 16:44:16 -0800
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--------------52F6C1A24B239DBE276F13F6
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: quoted-printable
On 2/15/2017 9:20 AM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:
>
> What an awesome puzzle and flurry of activity! This is really cool Melind=
a, and there is so much to think about. I especially find the limitations =
imposed by the magnets intriguing. I've been out of town and missed much o=
f the real-time fun, but read through the thread and watched Mathologer's v=
ideo last night. I must confess I have not fully digested all the great id=
eas flying around, but I had a few thoughts that still seem useful to share=
.
>
> Like Nan, my first instinct was to understand the lower dimensional case,=
and I'm glad I tried. I started to think of a 4x2 block of squares, and i=
t took me a bit to realize that was making it more difficult to think about=
. It may be obvious to folks, but my Aha moment was that each piece of Mel=
inda's puzzle is really a face of the dual polytope, the 16-cell n.wikipedia.org/wiki/16-cell>. In the lower dimensional case, each piece is=
a face of the octahedron. So it is natural to make the 2^3 analogue a set=
of triangles, a "net :Octahedron_flat.svg>" of them in fact. This all made me realize Melinda's=
16 pieces are representing tetrahedral faces, again a net pedia.org/wiki/16-cell#/media/File:16-cell_net.png> of the 16-cell.
>
> I think the brilliance of Melinda's design is that each tetrahedron is re=
presented by a cube, so the net becomes this nice 4x2x2 block. A less eleg=
ant way to approach this puzzle would be to use the net of 16 tetrahedra di=
rectly, which should work even if more awkward.
This is most clear when looking at Oskar's lovely rendering minal.com/cube/physical2x2x2x2oskar.jpg> of my previous concept. I was cons=
idering pieces as some sort of beads that would move along wire arcs, perha=
ps made out of some squishy material such as foam rubber. I don't remember =
an "aha" moment but the current design probably resulted from staring at th=
is picture and imagining various ways of squishing the parts while performi=
ng rotations.
> Using cubes for tetrahedra is possible because the tetrahedral group is a=
subgroup of the octahedral group. This doesn't work in the lower dimensio=
nal case because a triangle group is not a subgroup of a square group (look=
up cyclic groups if you want =
to research). That is why it was unnatural to deal with squares for the 2^3=
analogue.
So what does the 2D analog look like using triangles?
>
> So now I'm considering Melinda's puzzle as a net of the 16-cell, which ma=
kes it easier to think about what general rotations and twists are.
>
> * A rotation is any detachment of a connected subset of cells, and subs=
equent reattachment that preserves the structure of the net.
> * A twist is any planar cut of the net into two equal parts, followed b=
y an arbitrary reorientation of one of the halves (and optionally one could=
also make net preserving changes to the half too), then a reattachment bac=
k along the planar cut.
>
I like it!
> Given that, I thought I'd highlight a couple twists I don't think I've se=
en yet:
>
> 1. A slight modification of Melinda's reorienting move... Pull apart the=
two halves, but leave one fixed and rotate the second 90 degrees and reatt=
ach.
>
This is the same as a rotation followed by a simple 90 degree twist. The wa=
y I like to think about it is that you can pull the red-blue halves apart, =
reorient them, and connect them back together however you like. If the resu=
lt can be reached by a sequence of simple rolling moves, then it's a rotati=
on, otherwise it's a twist.
> 1. If we were to allow interim jumbling, I think we can get 90-degree tw=
ists of the blue and orange faces. Instead of performing a 180-degree rota=
tion maneuver here, you would take the 4x2 block and translate it a step. =
The end result would be a 3x2x2 with two 2x1 blocks protruding off of it on=
opposite sides. But then you could just do a reorientation by rolling one=
of those protrusions around to meet the other and recover the original 4x2=
x2 block. Hope this is clear.
>
I think the shearing step is clear enough though I'm not sure what orientat=
ion you intend the 2x1 block to end up in. Regardless, that would look like=
an even stranger result than anything I've seen so far because half of the=
pieces would have single stickers at the corners, and half would have 3-co=
lor junctions. I had been calling such oddly reoriented pieces "inverted", =
but is "jumbled" the more correct term? It would be nice if the terms jumbl=
ed and bandaged are the correct analogies.
> I want one Melinda! When are you going to set up a shop? :D
Well the first question is which version do you want? The bandaged (current=
) version or Matt's more general arrangement? I'm working with the dice guy=
right now to see if and how the 24 magnet pieces might be made and what th=
ey will cost. The magnets will likely need to be recessed so that it's not =
too hard to turn. That means either an extra step to fill in the gaps, or j=
ust sticker over them. The nice thing is that none of the diagonal sticker =
cuts will cross any magnets.
I'm happy to produce these for group members at cost, at least until it bec=
omes too much work. It took me nearly 4 hours to sticker this one (nearly 2=
00 triangles!) but I'm sure that with practice, I can soon do it in under 2=
hours.
-Melinda
>
>
>
>
> On Sun, Feb 12, 2017 at 9:35 PM, Melinda Green melinda@superliminal.com <=
mailto:melinda@superliminal.com> [4D_Cubing] <4D_Cubing@yahoogroups.com ilto:4D_Cubing@yahoogroups.com>> wrote:
>
>
>
> Yes, the reorientations were all 120 degree twists about the long dia=
gonals through their black stickers. I knew the black stickers had to remai=
n where they were because it was a twist of the black face.
>
> The magnets do not want to allow the twist or the piece reorientation=
s, though the central 2x2x2 cube was happy once it was fully reorientated. =
The frames where you see my hand are the configurations that the magnets do=
not allow. Matt's pattern should allow everything.
>
> -Melinda
>
> On 2/12/2017 7:23 PM, Christopher Locke project.eutopia@gmail.com ilto:project.eutopia@gmail.com> [4D_Cubing] wrote:
>>
>> Melinda,
>>
>> Yes, that looks like a correct twist of the +y hyperface about the +=
z axis! If the colors are labelled: (-x brown, +x purple, -y gray, +y blac=
k, -z light blue, +z green, -w blue, +w red), then that 90 degree twist sho=
uld move those middle 8 physical cubies around in a 90 degree twist just li=
ke you did, and the x/w stickers should move blue -> purple -> red -> brown=
. From step 3 to 4, I take it you did a 120 degree twist about a diagonal =
axis that goes through the center of each cubie and the total center of the=
physical puzzle (where black stickers are)?
>>
>> By the way, was the magnetic orientation okay after doing those twis=
ts? In the video move I pointed out (https://youtu.be/Asx653BGDWA?t=3D1410=
), you had problems doing some twis=
ts after the double inversion due to magnet positioning.
>>
>> Best regards,
>> Chris
>>
>> On 2017=E5=B9=B402=E6=9C=8812=E6=97=A5 17:55, Melinda Green melinda@=
superliminal.com [4D_Cubing] wrote:
>>>
>>> Matt,
>>>
>>> Christopher's messages were a bit to opaque to me but I'm starting =
to get the gist. It's good to know that this 90 degree twist is a valid mov=
e though it's unfortunate that it's far from pure. These almost look like g=
ear-cube twists now. I even think I can guess how the orientations are supp=
osed to end up after the appropriate reorientations of the black pieces. (A=
lternating CW and CCW twists of each piece about their black stickers.) I'v=
e attached a sequence of snaps showing the process. (Also here rliminal.com/misc/twist90cp.jpg> in case the attachment doesn't work.) The =
second snap shows the twist in progress. The third shows it completed, with=
me holding it in place against the magnets. You can see what I mean about =
the puzzle looking completely scrambled by this one twist. The fourth snap =
shows it with all 8 of the twisted pieces reoriented. The interesting thing=
is how it results in a much less scrambled looking puzzle.
>>>
>>> Christopher,
>>>
>>> I hope the photos helped. One interesting to note is that the end r=
esult of the sequence (plus a simple rotation) resembles the result of the =
double swap you highlighted in the video (https://youtu.be/Asx653BGDWA?t=3D=
1410 ) so maybe there's hope for a m=
ore practical way to reach the full 2^4 state space.
>>>
>>> Thanks!
>>> -Melinda
>>>
>>> On 2/12/2017 3:46 PM, damienturtle@hotmail.co.uk le@hotmail.co.uk> [4D_Cubing] wrote:
>>>
>>>
>>
>
>
>
>
>
>
>=20
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">
On 2/15/2017 9:20 AM, Roice Nelson " href=3D"mailto:roice3@gmail.com">roice3@gmail.com [4D_Cubing]
wrote:
cite=3D"mid:CAEMuGXrTc9p5PPP-KDdqe_nG6R6h2O9udL7A5_C5F7bKwzhmDg@mail.gmail.=
com"
type=3D"cite">
What an awesome puzzle and flurry of activity!=C2=A0
This is really cool Melinda, and there is so much to think
about.=C2=A0 I especially find the limitations imposed by the magne=
ts
intriguing.=C2=A0 I've been out of town and missed much of the
real-time fun, but read through the thread and watched
Mathologer's video last night.=C2=A0 I must confess I have not full=
y
digested all the great ideas flying around, but I had a few
thoughts that still seem useful to share.
I think the brilliance of Melinda's design is that each
tetrahedron is represented by a cube, so the net becomes this
nice 4x2x2 block.=C2=A0 A less elegant way to approach this puzzl=
e
would be to use the net of 16 tetrahedra directly, which
should work even if more awkward.=C2=A0
This is most clear when looking at href=3D"http://superliminal.com/cube/physical2x2x2x2oskar.jpg">Oskar'=
s
lovely rendering of my previous concept. I was considering
pieces as some sort of beads that would move along wire arcs,
perhaps made out of some squishy material such as foam rubber. I
don't remember an "aha" moment but the current design probably
resulted from staring at this picture and imagining various ways of
squishing the parts while performing rotations.
cite=3D"mid:CAEMuGXrTc9p5PPP-KDdqe_nG6R6h2O9udL7A5_C5F7bKwzhmDg@mail.gmail.=
com"
type=3D"cite">
Using cubes for tetrahedra is possible because the
tetrahedral group is a subgroup of the octahedral group.=C2=A0 Th=
is
doesn't work in the lower dimensional case because a triangle
group is not a subgroup of a square group (look up=C2=A0
moz-do-not-send=3D"true"
href=3D"https://en.wikipedia.org/wiki/Cyclic_group"
target=3D"_blank">cyclic groups if you want to research).=
=C2=A0
That is why it was unnatural to deal with squares for the 2^3
analogue.
So what does the 2D analog look like using triangles?
cite=3D"mid:CAEMuGXrTc9p5PPP-KDdqe_nG6R6h2O9udL7A5_C5F7bKwzhmDg@mail.gmail.=
com"
type=3D"cite">
So now I'm considering Melinda's puzzle as a net of the
16-cell, which makes it easier to think about what general
rotations and twists are. =C2=A0
- A rotation is any detachment of a connected subset of
cells, and subsequent reattachment that preserves the
structure of the net.=C2=A0=C2=A0
- A twist is any planar cut of the net into two equal
parts, followed by an arbitrary reorientation of one of
the halves (and optionally one could also make net
preserving changes to the half too), then a reattachment
back along the planar cut.
I like it!
cite=3D"mid:CAEMuGXrTc9p5PPP-KDdqe_nG6R6h2O9udL7A5_C5F7bKwzhmDg@mail.gmail.=
com"
type=3D"cite">
Given that, I thought I'd highlight a couple twists I don't
think I've seen yet:
- A slight modification of Melinda's reorienting move...=C2=
=A0
Pull apart the two halves, but leave one fixed and rotate
the second 90 degrees and reattach.
This is the same as a rotation followed by a simple 90 degree twist.
The way I like to think about it is that you can pull the red-blue
halves apart, reorient them, and connect them back together however
you like. If the result can be reached by a sequence of simple
rolling moves, then it's a rotation, otherwise it's a twist.
cite=3D"mid:CAEMuGXrTc9p5PPP-KDdqe_nG6R6h2O9udL7A5_C5F7bKwzhmDg@mail.gmail.=
com"
type=3D"cite">
- If we were to allow interim jumbling, I think we can get
90-degree twists of the blue and orange faces.=C2=A0 Instead =
of
performing=C2=A0a 180-degree=C2=A0rotation maneuver=C2=A0here=
, you would
take the 4x2 block and translate it a step.=C2=A0 The end
result would be a 3x2x2 with two 2x1 blocks protruding off
of it on opposite sides.=C2=A0 But then you could just do a
reorientation by rolling one of those protrusions around
to meet the other and recover the original 4x2x2 block.=C2=A0
Hope this is clear.
I think the shearing step is clear enough though I'm not sure what
orientation you intend the 2x1 block to end up in. Regardless, that
would look like an even stranger result than anything I've seen so
far because half of the pieces would have single stickers at the
corners, and half would have 3-color junctions. I had been calling
such oddly reoriented pieces "inverted", but is "jumbled" the more
correct term? It would be nice if the terms jumbled and bandaged are
the correct analogies.
cite=3D"mid:CAEMuGXrTc9p5PPP-KDdqe_nG6R6h2O9udL7A5_C5F7bKwzhmDg@mail.gmail.=
com"
type=3D"cite">
I want one Melinda! When are you going to set up a shop? :Div>
Well the first question is which version do you want? The bandaged
(current) version or Matt's more general arrangement? I'm working
with the dice guy right now to see if and how the 24 magnet pieces
might be made and what they will cost. The magnets will likely need
to be recessed so that it's not too hard to turn. That means either
an extra step to fill in the gaps, or just sticker over them. The
nice thing is that none of the diagonal sticker cuts will cross any
magnets.
I'm happy to produce these for group members at cost, at least until
it becomes too much work. It took me nearly 4 hours to sticker this
one (nearly 200 triangles!) but I'm sure that with practice, I can
soon do it in under 2 hours.
-Melinda
cite=3D"mid:CAEMuGXrTc9p5PPP-KDdqe_nG6R6h2O9udL7A5_C5F7bKwzhmDg@mail.gmail.=
com"
type=3D"cite">
=20=20=20=20=20=20
--------------52F6C1A24B239DBE276F13F6--
From: Roice Nelson <roice3@gmail.com>
Date: Thu, 16 Feb 2017 16:09:37 -0600
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
--f40304361cee9449a70548ad0d8c
Content-Type: text/plain; charset=UTF-8
>
>
> This is most clear when looking at Oskar's lovely rendering
> of my previous
> concept. I was considering pieces as some sort of beads that would move
> along wire arcs, perhaps made out of some squishy material such as foam
> rubber. I don't remember an "aha" moment but the current design probably
> resulted from staring at this picture and imagining various ways of
> squishing the parts while performing rotations.
>
>
Ah, thanks for sharing that again. I remember seeing it now, but had
forgotten!
> Using cubes for tetrahedra is possible because the tetrahedral group is a
> subgroup of the octahedral group. This doesn't work in the lower
> dimensional case because a triangle group is not a subgroup of a square
> group (look up cyclic groups
> if you want to research). That is why it was unnatural to deal with
> squares for the 2^3 analogue.
>
>
> So what does the 2D analog look like using triangles?
>
>
In my mind, it looks like a net of triangles of the octahedron (which can
take a number of jagged shapes
).
Each triangle has 3 lines that start at the triangle's edge midpoints and
meet at the triangle center. These define 3 colored areas, rather than the
4 colored volumes in yours. Each triangle would have 3 magnets, one at
each edge midpoint.
This analogue is not nearly as clean since the octahedron net is jagged.
> 1. If we were to allow interim jumbling, I think we can get 90-degree
> twists of the blue and orange faces. Instead of performing a
> 180-degree rotation maneuver here, you would take the 4x2 block and
> translate it a step. The end result would be a 3x2x2 with two 2x1 blocks
> protruding off of it on opposite sides. But then you could just do a
> reorientation by rolling one of those protrusions around to meet the other
> and recover the original 4x2x2 block. Hope this is clear.
>
>
> I think the shearing step is clear enough though I'm not sure what
> orientation you intend the 2x1 block to end up in. Regardless, that would
> look like an even stranger result than anything I've seen so far because
> half of the pieces would have single stickers at the corners, and half
> would have 3-color junctions. I had been calling such oddly reoriented
> pieces "inverted", but is "jumbled" the more correct term? It would be nice
> if the terms jumbled and bandaged are the correct analogies.
>
>
I'm using "jumbled" to mean something else, like this
. IOW, I
was using jumbled to mean allowing moves that don't result in the 4x2x2
block. Sounds like there are 3 separate terms here.
> I want one Melinda! When are you going to set up a shop? :D
>
>
> Well the first question is which version do you want? The bandaged
> (current) version or Matt's more general arrangement? I'm working with the
> dice guy right now to see if and how the 24 magnet pieces might be made and
> what they will cost. The magnets will likely need to be recessed so that
> it's not too hard to turn. That means either an extra step to fill in the
> gaps, or just sticker over them. The nice thing is that none of the
> diagonal sticker cuts will cross any magnets.
>
> I'm happy to produce these for group members at cost, at least until it
> becomes too much work. It took me nearly 4 hours to sticker this one
> (nearly 200 triangles!) but I'm sure that with practice, I can soon do it
> in under 2 hours.
>
I think I'd like to buy whatever version evolves from this thread that we
know will support the full 2^4 permutation group (though it would be cool
to have both). I'd be happy to sticker it myself. I recently bought a
Tuttminx that came unstickered,
and stickering it took me a similar amount of time.
Roice
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