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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