# Message #1388

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Interesting object

Date: Mon, 07 Feb 2011 23:02:37 -0600

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 tilings.org

table <http://tilings.org/pubs/tileclasstables.pdf>, 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<http://gravitation3d.com/magictile/8-3/8-3_tiling1.png> and

here <http://gravitation3d.com/magictile/8-3/8-3_tiling2.png>. 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<http://www.gravitation3d.com/magictile/pics/what_am_i.png>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 www.vihart.com, which I’m

sure many in this group would enjoy.

On Sun, Feb 6, 2011 at 8:20 PM, Melinda Green <melinda@superliminal.com>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 <http://superliminal.com/geometry/infinite/infinite.htm>.

> 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} <http://superliminal.com/geometry/infinite/3_7a.htm>), also known as

> Klein’s Quartic <http://math.ucr.edu/home/baez/klein.html>. 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<http://superliminal.com/geometry/infinite/infinite.htm>.

> 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<http://superliminal.com/geometry/infinite/3_8b.htm>which has an unusually high genus (five!). It is naturally modeled as a

> particular cubic packing of snub cubes<http://en.wikipedia.org/wiki/Snub_cube>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:

> http://physicsworld.com/cws/article/indepth/44950

>

> Hart’s in-depth page on the construct is here:

> http://www.georgehart.com/DC/index.html

>

> 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.

>

>