# Message #846

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Re: Introducing "MagicTile"

Date: Thu, 04 Feb 2010 22:29:45 -0800

Roice Nelson wrote:

>

>

> What an great flow of ideas. I love it!

>

> About solving towards a single cell, I think the program can help

> answer this by using the setting to only show the fundamental set of

> tiles. I’m not perfectly confident, but my intuition says it is

> possible to avoid multiple disjoint unsolved cells at the end. My

> reasoning is that even if the topology of the object as a whole

> (fundamental and orbit regions glued up) is not simply connected, the

> fundamental region alone is simply connected (as long as you never

> leave the boundary of it, so that it behaves topologically as a disk).

> Looking at only the fundamental set of Klein’s Quartic, it is easy to

> pick a solve order leaving only one cell at the end, by using sort of

> a "solve wave" to push unsolved cells from one end to the other. This

> would be my strategy to "keep the edge a simple closed curve",

> as Melinda nicely described things.

Can you solve a puzzle on a torus this way? It may be different than on

surfaces of higher higher genus but it’s a good first test of the idea.

>

> If I require that I solve cells in a connected fashion, where each

> cell solved must be adjacent to the prior one, it appears I lose the

> ability to end on a single cell. As an aside, I wonder if it is

> possible to have fundamental sets of tiles which are not simply

> connected, which could invalidate my logic above.

>

> Anyway, you’ve all convince me that it is the topology that is

> important factor in all this (though it still seems the symmetries and

> topology are intertwined). After taking another look at the

> 12-colored octagonal, also genus 3, I see it actually does behave the

> same.

>

> Regarding which puzzle(s) may be "just right", I’ve only done a few so

> far, but last night I did go through a solve of the 9-colored torus

> puzzle and can say I found it really enjoyable - perfect for a solve

> in one sitting for sure. And the repeating units caught me a couple

> times, which was fun. It may be inevitable that the 3x3x3 will always

> be the most perfect puzzle in my mind though :)

For me the 3^3 seems the hardest relative to it’s size but something

about the megaminx really appeals to me more these days. Maybe a little

too far in the tedious direction but I find the geometry very elegant.

> One small comment related to difficulty… The state-space of the

> 3-coloreds is small enough that they will often solve themselves if I

> just randomly click.

>

> Yesterday, I learned about some additional coloring possibilities on

> the hexagonal puzzles which I think will be good, balanced puzzles

> when it comes to the tradeoffs being discussed, e.g. a 7-colored (This

> was pointed out to me by someone from the twistypuzzles forums).

> Applying the tiling algorithm which produces those to the hyperbolic

> puzzles will probably allow some further balanced variants there as

> well. The 6-12 color range feels nice to me, both in terms of

> coupling and tedium.

>

> Btw, a while back I realized repeating units can be applied to

> spherical puzzles too, which will end up throwing even a few more into

> the mix. A Megaminx with a 6-colored fundamental region and opposite

> sides being orbits would topologically act as the non-orientable

> <http://en.wikipedia.org/wiki/Orientable> real projective plane

> <http://en.wikipedia.org/wiki/Real_projective_plane> (I verified by

> playing with color settings). Due to the non-orientability of that

> surface, orbits would rotate in an opposite sense to the fundamental

> cells! One could do a non-orientable hexagonal version as well,

> topologically a Klein bottle

> <http://en.wikipedia.org/wiki/Klein_bottle>. I definitely want to

> support these variants at some point, which will only involve minor

> code changes to deal with reversing the twisting.

Absolutely!

Non-orientable puzzles will work really well here. Their

self-intersections in physical models really detract from their beauty,

but in your presentation they’ll look extra cool as they twist. I’d love

to see Klein bottle puzzles and the even crazier Boy’s surface

<http://en.wikipedia.org/wiki/Boy%27s_surface>. BTW, if you do make a

megaminx with opposite faces identified, then it will have genus 6 which

should give solvers all kinds of conniptions. I don’t think that it

needs to be non-orientable, but either way it will be interesting to see

a high-genus puzzle. You should also be able to make twisty puzzles from

the duels of all the IRPs that I have listed in the first column of the

table on my IRP page

<http://www.superliminal.com/geometry/infinite/infinite.htm>. A

particularly interesting one there is a genus 5 {3,8}

<http://www.superliminal.com/geometry/infinite/3_8b.htm> made out of

snub cubes. It was one of the hardest models to work out the 3D vertex

positions for.

Remember, all of the suggestions that we’re making are because we’re

excited, and not because anything is broken or missing. I hope that you

don’t feel pressure to make additions just for us. Please take a

well-deserved rest and then come back and add puzzles and features when

you just can’t wait to see the results.

-Melinda