Message #2174

From: schuma <>
Subject: Re: Making a puzzle based on 11-cell
Date: Wed, 23 May 2012 02:50:37 -0000

Hi guys,

In the last week I’ve been working on the 11-cell. Although not completed, it’s a playable puzzle now. I’ll keep refining it. But I want to let you guys see the current version of the applet:

When you open it, you can see eleven hemi-icosahedral cells. The color scheme is exactly identical to the one here: []. But in that figure, the faces are colored to be identical to the opposite cell, whereas in my illustration, the faces are colored to be identical to the cell it belongs to. I also changed the layout of the cells to make it more intuitive.

There are 11 cells. I consider each cell has a core that never moves. The core is represented by a colored disk as the background of a cell. Just like the centers of the Rubik’s cube, the cores should be used as references during a solve. There are 2-color face pieces, 3-color edge pieces, and 6-color vertex pieces. When you move the cursor on a cell, the turning region will be highlighted. The region includes all the pieces that have stickers on the hemi-icosahedron.

Left or right clicking the mouse will twist the puzzle. I’m allowing twists around vertices and face centers. When the cursor is on top of a vertex, all the stickers on the vertex piece will be circled. When the cursor is on top of a triangular face, the two stickers on the face piece will be circled. These features can help you see the connectivity of 11-cell, to some extend. Please try click on the red cell, which is considered by myself as the "central cell". Twists around that cell are more symmetrical and thus understandable.

Holding shift and clicking mouse will re-orient the whole puzzle. Currently this is the only way to re-orient.

The checkboxes "F", "E", and "V" controls the visibility of the faces, edges and vertices. The percentages of stickers in the correct position are also shown. When all are solved, the background turns light blue.

The meaning of the buttons "reset" "scramble" and "undo" are self-explaining. I put a "1-move scramble" button for testing purpose. But I found it was already a nontrivial challenge. Let me know if you can solve it!

In the future I will fix bugs, and add macros. I haven’t seriously thought about how to solve it. But it looks like a pretty deep cut puzzle: each turn affects six out of eleven vertices, and it should be hard to solve. So I think macros should be necessary.


— In, Roice Nelson <roice3@…> wrote:
> 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<>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<>on
> the
> vzome <> 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 <> 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<>
> 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@…> 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 <
> >>. I found that this shape, together
> > with the more complicated 57-cell, was mentioned once in our group in this
> > post <> 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 <
> >> 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: <
> >>. 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 <
> >> 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
> >
> >
> >
> >