Message #1733

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] puzzle avalanche continues
Date: Sun, 22 May 2011 00:46:17 -0500

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

Can you send me an image of the bumping you are seeing? I uploaded a pic of
a {5,4} <http://www.gravitation3d.com/magictile/pics/%7B5,4%7D.png> as it is
rendering for me, and all the ultra-parallel lines of cell boundaries appear
as nice, smooth arcs.

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 I’ll play with this some. I’ve since realized a puzzle based on
this coloring may still be possible… if the twisting circles are smaller
than the circumcircle for a face, we can keep them from intersecting.

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

Done, and uploaded<http://www.gravitation3d.com/magictile/downloads/MagicTile_v2_Preview.zip>
:)
Naming is something I wish I was more creative with, so if anybody else is
struck by names they like, please let me know!

I probably should wait longer before mentioning this (until things are more
stable), but if anyone would like to try to make their own puzzles, you can
copy some of the existing puzzles in the config/puzzles directory to the
config/user directory to use as a template, then edit them. They will then
show up in the menu, and if you create any good ones, you can send them to
me to include in the standard list of puzzles. The display name is just one
of the xml nodes, so you can be free to give your creations any unique name
you’d like. Strange configurations can easily make the program puke, which
is why this suggestion is probably premature. I’ll plan to do a round to
make failures in this area more robust, but will throw caution to the wind
in the mean time. Just be warned :)

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.
>

Awesome, thanks! The great thing about this is that with the "show only
fundamental" setting, counting these elements is greatly simplified. In
fact, with your description, I may very well be able to automate the genus
calculation in code :)

Take Care,
Roice