Message #1389

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: [MC4D] Interesting object
Date: Tue, 08 Feb 2011 09:05:23 -0000

Roice,
Yes, their picture of the vertex shows that there are really 8 triagles meet at the point, but they call it (10,3) lattice… It looks like the surface of the periodic subset of octahedral-tetrahedral lattice. May be you know how to describe this subset?
My method of painting of (8,3) gives 28-color painting! Picture is here: http://astr73.narod.ru/pics/8x3_28.jpg Cells are indexed by frations a/b, where a,b are from Z/8Z, and fractions a/b and (-a)/(-b) are equivalent, but a/b and (3a)/(3b) are different. If cells a/b, c/d and e/f have common vertex, then a+p*c+q*e=0, b+p*d+q*f=0 for some p,q=1 or -1.
So we have 28 colors: 1/0,3/0,n/1,n/3,k/2,k/6,1/4,3/4, where n=0..7, k=1,3,5,7. If we unify colors a/b and 3a/3b, we’ll get non-orientable 14-color puzzle.
And about your riddle - it is too simple for me. Is it OK to leave it to somebody else?

Good luck!
Andrey

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> 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 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
>