# Message #1386

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Interesting object

Date: Sun, 06 Feb 2011 18:20:19 -0800

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.

>

>