Message #1392

From: Melinda Green <>
Subject: Re: [MC4D] Interesting object
Date: Tue, 08 Feb 2011 03:13:49 -0800

Regarding the 24 coloring of the {8,3}, I don’t think that the
hyperbolic view will help to visualize this very much because of the
extreme curvature, but I think that I can describe it fairly well. The
trick is to look at my {3,8}
<> screen shot and
hold the idea in your head that there is really only one red sphere and
one green one. The rest are just copies. So the repeat unit is this
two-sphere peanut shape. The octahedra are centered on each of the
vertices, with edge length 1/2, That is, pairs of octahedra meet at the
midpoints of each edge. Each octahedron consists of 4 green triangles
and 4 red ones folded into a sort of floppy butterfly shape. Since each
octahedron can be uniquely identified by the vertex at its center, the
coloring problem boils down to coloring each vertex of one faceted peanut.

Now notice that each vertex is part of a ring of 4 vertices surrounding
the squares where pairs of snub cubes meet. This says that all octagons
will participate in a exactly one ring of 4 octagons, stacked like 4
stop signs on the same pole. These will create straight parallel lines
across the hyperbolic plane. We can color these 1, 2, 3, 4, 1, 2, 3,
4,… Also, notice that to get from one ring to a copy of that ring, you
will have to first encounter another parallel ring which is on the
opposite side of a snub cube. Color those 5, 6, 7, 8, 5, 6, 7, 8,…
This pattern then repeats, alternating between parallel lines of the
first four colors with lines of the second four.

Since a cell of any given color now has two neighbor cells colored, that
accounts for 1/3 of all the cells. Another way to look at that is that
we’ve accounted for a stack of alternating red-green spheres in one of
the 3D coordinate axes. There are two more axes to go, accounting for
all 24 colors. In the hyperbolic plane, those lines neither parallel to
the first set, nor do they cross them, which I find *very* interesting!
I’ve numbered the coloring of one of Roice’s nets here
There are still some more large cells to fill in but I’m too tired to
try to fill in more. Hopefully this should give enough to show the pattern.

Regarding puzzles made from the {3,7} and {3,8} directly, that would
definitely be neat. It may even help to be doing this in the plane in
order to understand the general problem of non-simplex vertex figures.
What I’d *really* prefer would be to see those puzzles as painted onto
the surface of either repeating polyhedral models or onto curved minimal
surfaces. I can easily supply the coordinates and connectivity data for
the polyhedral models if anyone wants to try that. The animations could
be accomplished by animating textures mapped to those surfaces. That
would be very fascinating to see.

As for your mystery puzzle, it looks to me like a simple planar tiling
of hexagons inverted across a central circle. Did I get that right?


On 2/7/2011 9:02 PM, Roice Nelson wrote:
> Interesting… so this means there should be an alternate, genus 5
> puzzle based on {8,3}. And when I looked at genus 5 tilings at the
> table <>,
> indeed it suggests there should be a 24-color version. It must have
> some tiling pattern which does not fit into the simple rule I’ve used
> for the current puzzles. I’ll plan to investigate at some point, but
> it would be awesome if someone wanted to see if they can figure out
> the pattern of 24 colors which fit together, then explain it to me :)
> I made some blank pictures of an {8,3} tiling, hopefully suitable for
> printing, to help out any who are interested in tackling the problem.
> They are here
> <> and here
> <>. Thanks for
> sharing the associated IRP Melinda - that arrangement of snub cubes
> really is beautiful.
> I’d love for MagicTile to handle the {3,7} and {3,8} triangle-faced
> puzzles directly too, rather than just their duals, and have started
> investigating how these might be sliced up. Andrey’s amazing
> observation of edge behavior on Alex’s {4,4} puzzle made me wonder
> what else all the puzzles with non-simplex vertex figures have in
> store for us!
> By the way, anyone want to make a guess as to what this puzzle
> <> is?
> (hint: it is in conformal disguise)
> Cheers,
> Roice
> P.S. I got to meet George at the Gathering for Gardner conference last
> March. His daughter Vi was there as well, and has been making some
> waves online of late. She has a unique site at
> <>, which I’m sure many in this group would enjoy.
> On Sun, Feb 6, 2011 at 8:20 PM, Melinda Green
> < <>> wrote:
> Yes, that’s definitely an interesting object, and yes, it does
> relate to our particular interest. First, I think that George Hart
> is slightly obscuring what I feel is the more natural way of
> describing the polyhedron by having some edges crossing shared
> verticies as opposed to terminating there. To simplify this
> unusual construction, just subdivide each of those big triangles
> into four smaller ones and then the object is much more easily
> described. That version also appears to be missing from my
> collection of infinite regular polyhedra
> <>. George
> Hart helped me with these IRP’s by copying an out-of-print book
> with a collection of many figures containing many that I didn’t
> already know about. Even in its subdivided form, this polyhedron
> appears to be new to me and not the book.
> BTW, I know that George found my collection interesting because he
> once copied my VRML files for the {5,5} which I had painfully
> worked out on my own, and then he hosted it on his site without
> attribution, even carefully removing my name from the comments. At
> least he took it down when I confronted him. He’s been a very nice
> and enthusiastic booster of highly symmetric geometry and a
> generally nice and brilliant guy.
> The way that his new surface relates to twisty puzzles is exactly
> the same way that Roice implemented the twisty version of the
> {7,3} (duel of the {3,7}
> <>), also known
> as Klein’s Quartic <>. Any
> of these sorts of finite hyperbolic polyhedra that live in
> infinitely repeating 3-spaces (and probably many more that don’t)
> can be turned into similar twisty puzzles, especially ones in
> which 3 polygons meet at each vertex. All of the possibilities
> that I know of would be the duels of any of the polyhedra in the
> "Triangles" column of the table on my IRP page
> <>. George’s
> gyrangle, once subdivided as I described above, can be seen as an
> infinite {8,3}, and it’s {3,8} duel could be made into a puzzle.
> Another particularly beautiful {3,8} is this one
> <> which has an
> unusually high genus (five!). It is naturally modeled as a
> particular cubic packing of snub cubes
> <> meeting at their square
> faces, and with those faces removed. It was also the hardest of
> all the IRP for me to model as there does not seem to be a
> closed-form solution with which to compute the vertex coordinates.
> Don helped me out with a method of computing them with an
> iterative function to compute the coordinates to any required
> accuracy. I think that you will agree from the screen shot that
> the 3D form is particularly beautiful. I have no idea how
> difficult the resulting planar puzzle might be but I’d definitely
> love to see it implemented. I’m looking at you, Roice. :-)
> Thanks for reporting on this new object, David. It’s definitely
> interesting and pertinent in several ways.
> -Melinda
> On 2/6/2011 4:48 PM, David Vanderschel wrote:
>> I just read the following article:
>> Hart’s in-depth page on the construct is here:
>> Though I have not completely groked it yet, it struck me that
>> there might be yet another opportunity for a permutation puzzle
>> here; so I am curious to see what insights some of the folks on
>> the 4D_Cubing list might have. It would not surprise me if a
>> connection can be found with some objects which have been
>> discussed here.
>> Regards,
>> David V.