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 Post subject: Re: visualization of 8th cell for 4D Rubik's cube
PostPosted: Tue Jul 15, 2014 3:33 pm 
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wwwmwww wrote:
I've really enjoyed this thread. I have a question but to ask it properly I need to verify a few things first.

Lets start with a 2D Rubik's Cube.

Shape: Square
Faces: Lines (4 of them)
Stickers: Lines (3 on each face)
Turns available: 4 A rotation about the axis through the center of each face.

You broke the analogy here. You bestowed more properties on the 2D 3x3 than it wold actually have. Here is what it should look like:
Attachment:
220px-2-cube.png
220px-2-cube.png [ 2.69 KiB | Viewed 68 times ]


Because this is a 2D puzzle, only rotations in 2D are allowed which restricts this to rotating in the plane along the axis pointing out of the screen. It only has 4 orientations and can't be scrambled.

If instead you allowed it to have the two axes through the face centers like you describe it would be a 2D puzzle rotating in 3D and the rotations would act like mirror operations in 2D. Yes this is more interesting but it just doesn't match the properties of the 3x3x3 cube.


wwwmwww wrote:
The 3D Rubik's Cube.

Shape: Cube
Faces: Squares (6 of them)
Stickers: Squares (9 on each face)
Turns available: 6 A rotation about the axis through the center of each face.

Why are there not 24 turns available? Each of the 6 faces can be in 4 positions?

wwwmwww wrote:
The 4D Rubik's Cube

Shape: Hypercube
Faces: Cubes (8 of them)
Stickers: Cubes (27 on each face)
Turns available: 24? It appears each face can be rotated on 3 axes and there are 8 faces so that is where I get this number.
Actually 192 turns are available. There are 8 cells, each cell is a cube with 24 orientations. MC4D gives you 1-click access to 21 of them but you have to use 2-clicks to reach the "half turn" about a face twists.

wwwmwww wrote:
So I'm confused now by the 2D Rubik's cube. Do I understand the turns available correctly? It appears that a rotation requires a face to go out of the plane so maybe it is more a mirroring operation than a rotation. And I'm not sure I'm correct about the 24 turns available on the 4D Rubik's cube. Having multiple axes of rotation through a single face doesn't seem like an expected generalization from 3D to 4D? There wasn't an increase from 2D to 3D. How many turns are available on a 5D Rubik's Cube. If each face becomes a Hypercube and the trend appears to say there would be 10 faces how many turns are possible? Guessing it may be 240. 24 axes for each Hypercube and 10 faces total. Is that correct?


I think I addressed most of these above. I suppose there will be some debate about that though :) .

For the 3^5 there are 10 4-D tesseract cells each with 192 turns available (for a total of 1920 turns).

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 Post subject: Re: visualization of 8th cell for 4D Rubik's cube
PostPosted: Tue Jul 15, 2014 4:39 pm 
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Brandon Enright wrote:
Why are there not 24 turns available? Each of the 6 faces can be in 4 positions?
I was just thinking of the type of turns. i.e. a rotation of the R face about the axis through that face. All 4 positions that can be reached are reached with what I'd count as one type of turn. Maybe not the most formal but that is what I was thinking. No need to differentiate between clockwise and counter clockwise as both take you to the same positions.

Carl

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 Post subject: Re: visualization of 8th cell for 4D Rubik's cube
PostPosted: Tue Jul 15, 2014 4:54 pm 
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wwwmwww wrote:
Brandon Enright wrote:
Why are there not 24 turns available? Each of the 6 faces can be in 4 positions?
I was just thinking of the type of turns. i.e. a rotation of the R face about the axis through that face. All 4 positions that can be reached are reached with what I'd count as one type of turn. Maybe not the most formal but that is what I was thinking. No need to differentiate between clockwise and counter clockwise as both take you to the same positions.

Fair enough. I think this minimalist approach makes a lot more sense for 3D than 4D though. On a Rubik's cube each face has only 4 rotation states and it's easy for us to understand how just a clockwise turn can reach them all so saying only "one turn type" is available makes sense even though that one turn reaches 4 states.

On the 4D tesseract each cell is a cube which has 24 rotations available to it. You only need two turn types to reach them all though. So imagine turning the cube by the U face and then R face. Doing RUUU twists the whole cube about the UFR corner. Doing RUR twists the whole cube about the UR edge. There are 24 unique states that can be reached but now the order of operations matters and there isn't a unique sequences of twists to reach all 24. I think it just makes more sense to say there are "24 twists available per cell" and not try to use any shorthand like "two turns types per cell".

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 Post subject: Re: visualization of 8th cell for 4D Rubik's cube
PostPosted: Wed Jul 16, 2014 10:14 am 
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This is just my view and maybe not correct mathematically or puzzlogically.

If an axis is a set of points invariant by the rotation, an axis of rotation of 4D cube forms a plane(a 2D entity).

Ignoring cw or ccw, there are 3 90° rotations about axes perpendicular to faces of a cube
Code:
f(x,y,z) = (-y, x, z) # about z
f(x,y,z) = ( x,-z, y) # about x
f(x,y,z) = ( z, y,-x) # about y

Ignoring cw or ccw, there are 6 90° rotations about axes perpendicular to cells of a tesseract :? :? :?
Code:
f(w,x,y,z) = (-x ,w, y, z) # about yz-plane
f(w,x,y,z) = ( w,-y, x, z) # about zw-plane
f(w,x,y,z) = ( w, x,-z, y) # about wx-plane
f(w,x,y,z) = ( z, x, y,-w) # about xy-plane
f(w,x,y,z) = ( w,-z, y, x) # about wy-plane
f(w,x,y,z) = ( y, x,-w, z) # about xz-plane

On my 8.1.1(2x2x2x2), clicking on two side of a face(entity connecting 2 cells) are same rotations but work on different layers.
So doing them to all layers result to the same orientation of the whole tesseract.
for example:
Code:
AG',//axis : wx , layer: z>=0
HB',//axis : wx , layer: z< 0

GA, //axis : wx , layer: y>=0
BH, //axis : wx , layer: y< 0


Another comparison in terms of group :

Orientations of a cube form a group with 24 elements divided into 5 conjugacy classes.
Code:
--+-------------------+-----
1 | Identity          |  1
2 | 90° face turns    |  6
3 | 180° face turns   |  3
4 | 120° vertex turns |  8
5 | 180° edge turns   |  6
--+-------------------+-----
  | total             | 24
--+-------------------+-----

Orientations of a tesseract form a group with 192 elements divided into 13 conjugacy classes.
Code:
--+-------------------+-----
1| Identity          :  1
2| Center            :  1
3| modulo 2          :  6
4| modulo 4          :  6
5| modulo 4          :  6
6| modulo 2          : 12
7| modulo 4          : 12
8| modulo 4          : 12
9| modulo 2          : 24
10| modulo 8          : 24
11| modulo 8          : 24
12| modulo 3          : 32
13| modulo 6          : 32
--+-------------------+-----
  | total             |192
--+-------------------+-----

They include classes corresponding to face,vertex and edge turns of cube but others too.
Which interests me particularly is modulo 6 and modulo 8 operations. Theoretically it's possible to make puzzles with only these operations(maybe they already exist?) But I don't know how to describe them in words or in analogy with 3D. For example this operation has a modulo8-cycle.
Code:
f(w,x,y,z) = (-z,w,x,y)

What remains invariant ? Or how can I describe the equation of the axis ?
I have no idea.

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