Message #1732

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] puzzle avalanche continues
Date: Sat, 21 May 2011 18:36:14 -0700

Wow, what a bounty of riches!

The ultraparallel lines are indeed beautiful. Can the edges be adjusted
so that those lines are straight rather than bumping along?

Still haven’t figured out the 8-color {5,5}. I don’t know how pretty or
interesting it may turn out to be but it is definitely close to my
heart, topologically at least.

I think that my favorite is the 8-colored Euclidean {6,3}. It is easy to
grok and will look very familiar to all twisty puzzle enthusiasts while
having its exotic non-orientableness front and center.

I would like to name your 9-color edge turning {4,4} to be the
"Harlequin" tiling.

Regarding calculating genus, it is not difficult though you do have to
be extremely cautious in your counting. You need to count the number of
*unique* vertices, edges and faces in a single minimal repeat unit and
plug those values into the Euler formula F-E+V = 2-2g and solve for g.
Just go super slow so that you don’t skip any unique elements or count
any more than once. For instance, a simple toroidal {4,4} repeat unit is
a simple open cylinder with exactly 4 vertices, 4 horizontal and 4
vertical edges, and 4 faces. Plugging into the Euler formula you get 4 -
8 + 4 = 2 - 2g. Solving for g we get g = (0- 2)/-2 = 1 which is what we
would expect for any torus. See here
<http://superliminal.com/geometry/infinite/infinite.htm> for the
complete description with diagrams.

-Melinda

On 5/21/2011 12:55 PM, Roice Nelson wrote:
>
>
> I have yet to solve any, as I’m still getting distracted with finding
> new puzzles and colorings :) If any of the following sound fun, grab
> the latest here
> <http://www.gravitation3d.com/magictile/downloads/MagicTile_v2_Preview.zip>.
>
> * Two new {5,4} Petals, a non-orientable 6 color and an orientable
> 12 color (the relationship between the two is reminiscent of
> that between the Megaminx and hemi-Megaminx). The first is like
> an easier version of the {5,5} Petal, because there are no 1C
> pieces. Although these puzzles feel pretty simple and only have
> 2C edge pieces, both are as deepcut as can be, since the
> twisting circles are tangent to their identified counterparts.
> Also, I’m liking the hyperbolic tilings with square vertex
> figures because you get beautiful ultra-parallel lines running
> everywhere.
> * A 5 Color {4,5} with a neat slicing, but I bet it will be pretty
> difficult to solve. This one is also as deepcut as can be, but
> in this case that’s deeper than with the {5,4}s above.
> * A Pretty {3,6} with 8 Colors. This torus puzzle should be a fun
> one.
> * {4,4} Edge Turning, 9 Colors.
> * Yet another {8,3}, with 8 Colors this time. There are so many
> possible coloring combinations, and wild how some end up fitting
> together! I should figure out how to calculate the genus of all
> these guys.
>
> Cheers,
> Roice