# 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