# Message #266

From: Roice Nelson <roice@gravitation3d.com>

Subject: other polychora and polytope permutation puzzles

Date: Sat, 03 Jun 2006 15:35:00 -0500

I had the thought the other day that it would be cool to do a 4D version of

the Megaminx. A little wikipedia and web searching revealed that the word

‘polychoron’ is a common (though not standard) word for a 4D ‘polytope’,

which is the word for a generalized d-dimension polygon (a 3D polytope is a

polyhedron). Also, I found that the 120-cell (

http://en.wikipedia.org/wiki/120-cell) is the polychoron that could be

considered the 4D analog of the dodecahedron, which the Megaminx is based

on.

There is a really cool picture of an exploded 120-cell projected into 3D

space about halfway down the page at

http://home.inreach.com/rtowle/Polytopes/polytope.html. This is mostly what

I was thinking about, though I imagined slightly different projection

parameters that would have the closest face hidden like in MC4D. I saw a

picture having projection parameters I like at

http://www.bathsheba.com/math/120cell/, though it isn’t a cool exploded

view. The latter is more analogous to how I tried to flatten a 3D

dodecahedron into 2D when thinking about this.

Maybe the Magic120Cell puzzle would be too much with 120 faces. If

they couldn’t realistically be distinguished by colors, stickers might have

to be numbered. I have seen a Megaminx with only 6 colors which had colors

repeated on opposite sides. Maybe doing something like that could help,

though it does change the nature of the puzzle (aside: I wonder if repeating

colors, but not on opposite sides, could produce a Megaminx that behaved

exactly as the 12 colored one, since the 1-colored center pieces can’t

change position and the other pieces have multiple colors to help

distinguish where they belong. I’ll have to think more on that. If it was

the case, maybe colors on a 120cell could even be repeated more than once

without changing the puzzle behavior.?.)

Anyway, if the 120-cell is over the top, other polychora could make good

puzzles. The 4-simplex (http://en.wikipedia.org/wiki/4-simplex) would be

the 4D analog of Pyraminx, and would be easier and less screen-crazy than

MC4D. That puzzle would also be interesting because as I imagine one

projection of it, there would be no center face. There would be 4

icosahedrons all pointing towards an empty center. The 5th icosahedron

would be the hidden face closest to the viewer (think of the 3D to 2D case

of projecting the icosahedron of a Pyraminx to 2D).

An interesting thing about these puzzles is that relative to the orthogonal

coordinate axes, face rotations can affect all coordinates of a sticker.

Just like some face twists on Megaminx can change all 3 coordinates of the

moved stickers, some face twists on Magic120Cell or Magic4Simplex could

change all 4 sticker coordinates. This is unlike the cube puzzles, and it

seems this could make the programming more difficult, e.g. in MC4D you can

reduce face rotations to 3D rotations by simply not dealing with 1 of the 4

coordinates, but this will no longer be possible. This wasn’t a problem in

MC5D either, because we limited rotations to those parallel to coordinate

planes and all face rotations only changed 2 coordinate values for

stickers. I don’t understand it yet, but apparently 2 quaternions can

represent any general 4D rotation (vs. 1 that is required for 3D

rotations). What I don’t know is how to interpret the quaternion parameters

with a geometric meaning, like the rotation plane and rotation angle. For a

3D rotation, the quaternion that represents it has a direct geometrical way

to think about it in terms of an axis of rotation and an angle, but I’m

not sure if the 4D case even has a similar geometrical interpretation. If

anyone has any information on this, I would appreciate it.

Roice