Message #1386

From: Melinda Green <>
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
<>. 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.

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.