# Message #1734

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] puzzle avalanche continues

Date: Sun, 22 May 2011 01:57:50 -0700

On 5/21/2011 10:46 PM, Roice Nelson wrote:

>

> 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.

Now I’m not sure which puzzle I was commenting on. Looking again, the 6

color {5,5} is the only "petals" puzzle that look bumpy. All the rest

are 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 :)

Yea! I’m just glad that I had something mathematical to contribute.

Having the software do the arithmetic is a great idea because it is *so*

easy to mess up when attempting it by hand. Before I got the above right

I made the classic mistake of dropping a minus sign and ended up asking

myself for about the hundredth time whether a surface with a negative

genus makes any sense. (It doesn’t)

Another possible way to screw up is to choose a repeat unit that is not

minimal. Sometimes it’s easier to build a tiled surface from some

multiple of minimal repeat units, but you can’t use them to compute the

genus of the surface. So it’s very possible for someone builds a puzzle

that way causing your code to produce the wrong genus. I guess that any

way you do it you still need to be very careful. Be warned indeed! :-)

-Melinda