Message #2159

From: schuma <mananself@gmail.com>
Subject: Making a puzzle based on 11-cell
Date: Mon, 14 May 2012 20:06:36 -0000

Hi everyone,

Last night I was surfing the internet looking for some potential shapes to make puzzles. Then I looked at the 11-cell <http://en.wikipedia.org/wiki/11-cell>. I found that this shape, together with the more complicated 57-cell, was mentioned once in our group in this post <http://games.groups.yahoo.com/group/4D_Cubing/message/1320> and Andrey’s reply. But there’s no follow up discussion about them.

11-cell is an abstract regular four-dimensional polytope, where each cell is a hemi-icosahedron. It can be illustrated nicely in this way <http://en.wikipedia.org/wiki/File:Hemi-icosahedron_coloured.svg> by drawing eleven half icosahedra, with many vertices, edges and faces identified.

I feel that we can use this illustration as the interface of the "Magic 11-cell" simulator. We can define the following pieces: eleven 6-color vertices, fifty-five 3-color edges, fifty-five 2-color faces, and eleven 1-color cell centers (never move). We ignore other tiny pieces that would come if we define a proper geometry in a high dimensional space and properly cut the puzzle by hyperplanes (think of the small pieces in the 16-cell puzzle).

After defining the pieces, we can consider a twist of a hemi-icosahedral cell as a permutation of the vertices, edges and faces related to that cell. After that, a cell-turning 11-cell will be well-defined.

Through the wikipedia page, I found some recent presentations by Prof. Carlo Sequin at UC Berkeley about visualizing the 11-cell and the 57-cell. Since I’m also at Berkeley and there’s his office hour this morning, I stopped by his office, introduced myself and discussed happily about the possibility of combining a twisty puzzle with the 11-cell. He confirmed this possibility and was happy to see it coming out someday.

About the puzzle based on a single hemi-icosahedron, which has been implemented in Magic Tile v2, he suggested a 3D visualization based on this octahedral shape: see Fig. 4(c) in this paper: <http://www.cs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf>. He likes this shape a lot. While it’s a good visualization for one hemi-icosahedron, it’s hard to imagine combining eleven of them to form a 11-cell. So he said the illustration <http://en.wikipedia.org/wiki/File:Hemi-icosahedron_coloured.svg> would be better for the 11-cell.

It turns out he knew Melinda Green through the Gathering for Gardner meeting. And he said that he had some thoughts about using the 11-cell as a building block of IRP, and he needs to write to Melinda about it.

His office is filled by tons of different Math models, paper-made or 3D printed. It’s like a toy store.

Any thoughts for this 11-cell thing?

Nan