# Message #2160

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Making a puzzle based on 11-cell

Date: Mon, 14 May 2012 20:24:31 -0500

Hi Nan,

I have devoted a little thought to the idea of making puzzles out of the

11-cell and 57-cell, mainly the latter. I also would love to see a

realization of these abstract polytopes as puzzles.

For the 57-cell, it turns out you can consider it derived from the {5,3,5}

hyperbolic honeycomb<http://en.wikipedia.org/wiki/Order-5_dodecahedral_honeycomb>with

identification of certain elements (wiki says this anyway). Because

the cells themselves are abstract, it may be more complex than just

identification of cells like in Andrey’s MHT633. But there could be a nice

representation achievable by taking the approach of showing the object

unrolled on the {5,3,5} honeycomb.

As far as showing the object rolled up, my best mental image so far has

been to display an embedding the graph with some kind of coloring attached

to the edges. There would be no solid 3D stickers, just edges with small

ribbons of colors coming off of them (like Sequin’s images of the 11-cell

in the paper you link to). For the 57-cell, each edge would have 5

attached colors. Twisting would break some edges apart and connect them

back together to others at the twist end. If animated, the movements would

involve all kinds of distortions. I hope I’m painting the mental picture

well enough.

Toward that end, I’ve played around with embedding the graph of the 57-cell

in 3D space, in the attempt to find nice ones. Since the object lives

in such a high dimensional space, I’ve had little success. But I can tell

you how to connect up the graph (and can share code on this if anyone was

interested). There is an open

challenge<http://math.stackexchange.com/questions/69180/can-the-57-cell-be-made-in-vzome-without-strut-crossings>on

the

vzome <http://www.vzome.com> group to find a 57-cell skeleton model in zome

using a limited set of directions and without intersection of graph edges.

We don’t know if it is possible, but after my attempts I’m confident saying

it will look pretty ugly if such a model is found! We’ve found many nice

zome embeddings of the hemi-dodec though. And without the restrictions of

zome, there are probably some reasonable looking embeddings that could be

used for a puzzle.

These thoughts can apply to the 11-cell as well. Maybe the icosahedral

honeycomb <http://en.wikipedia.org/wiki/Icosahedral_honeycomb> be used with

identification of elements (?) The "graph with small colored ribbons"

approach seems like it would work better in this case because the

graph embedding is less complex.

For the wiki image you link to, if the puzzle representation were based on

this, I think it would be nice if the pristine state showed solid colored

hemi-icosahedra rather than multicolored ones. They look to be trying to

show all the connections between cells in the wiki image, but having them

multi-colored makes it feel like a 2D puzzle, when it is so far from that :)

Although any representation would be an achievement, I’m heavily biased

towards those which are connected myself. Even if connected versions are

messy looking on the screen, I find the dissected variants less elegant.

(With MC5D, I was never willing to approach it by showing the hyperfaces

laid out side by side. I wanted it all connected up.)

Anyway, those are some quick thoughts, but I’m interested to discuss and

spec more on these abstract puzzles!

Cheers,

Roice

P.S. I was able to visit with Carlo a little at the Gathering as well, and

really enjoyed the brief time I got to talk with him. He thinks about

really amazing things, and I just love hearing what he has to say. His

talk was on the 11-cell. Here’s a short

paper<http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf>

he

wrote on the 57-cell. It’s cool you have access to his office hours.

On Mon, May 14, 2012 at 3:06 PM, schuma <mananself@gmail.com> wrote:

> 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

>

>

>

> ————————————

>

> Yahoo! Groups Links

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>

>