Message #1450
From: Roice Nelson <roice3@gmail.com>
Subject: slicing up MagicTile puzzles without triangle vertex figures
Date: Tue, 22 Feb 2011 22:18:57 0600
Hi all,
I made a toy to help study the problem of how to slice up (face turning)
MagicTile puzzles that do not have triangle vertex figures, and wanted to
share. Honestly, my initial impression is that I wish the slicing turned
out to be more elegant in the general case. Instead, there seem to be a
huge number of possible puzzles for tilings like the {3,7}, none of them
which feel particularly natural to me. You can play with the study tool
directly in a web browser if you have Silverlight installed (or are willing
to install it). I seem to be overtaxing the Silverlight drawing a bit, and
some of the spherical puzzles aren’t perfect due to things projecting to
infinity, but it serves the purpose I wanted pretty well.
www.gravitation3d.com/magictile/slicing_study.html
Here are a few thoughts I had, but I’d really love other opinions on what
would be the best puzzles for the next iteration of MagicTile.

Starting with a small circle size and increasing, the transition between
puzzle types happens at points where new intersections begin between sets of
two or more slicing circles (it reminds me of Venn diagrams). All the
possible ways in which this can happen are very complicated. As you
increase the circle size, there can be *a lot* of puzzle "phase
transitions". 
The puzzles with square vertex figures seem to work pretty nicely, in that
it is very easy to slice cells into 3perside, but once you get up to
pentagonal vertex figures, it feels like things are getting messy. 
I could not see a way to slice the {3,7} into 3perside if the slicing
circles get larger than the default (which is passing through the vertices
of their parent cell). Once the circles are slightly bigger than that, the
tiling slices into 7perside! You can reduce that to 5 at certain points. 
You can slice up any of the puzzles into 3perside if you make the
slicing circles smaller than the default (such that they just move beyond
the parent cell edges but have not yet reached the vertices). These
actually seem like a nice class of puzzles to support, even if they’ll be
easy. They include puzzles where the only pieces you can permute are 2C
edges, but also some other types where centers and/or interior pieces can
permute around. 
On the {3,6}, if you make the circles larger than the parent cell, you can
slice into 3perside by making the slicing circle go 2/3rds the way across
some adjacent cell edges (which simultaneously puts it 1/3rd the way across
some incident cell edges). This feels like a nice puzzle to me, with a
pretty star pattern in the middle of each cell. You can do the same thing
on the {3,5} icosahedron, but in that case, the cuts are not evenly spaced
along an edge and the star patten is not quite as regular. You can also do
it on the selfdual {5,5}, and the resulting pattern is very wild  that
one looks like an extremely difficult puzzle. 
There is a very cool midpoint slicing of the {3,4}. The doubledup
slicing circles form a cuboctahedron, so this is a case where things do fit
together quite nicely.
I can capture pictures of some of the above if people need, but hopefully
you are able to play with it directly to follow what I’ve written. I
couldn’t picture this stuff very well ahead of time myself, without seeing
it in motion.
Anyway, I’m curious what puzzles people would like to see actualized, if
any. Are there any slicing options which feel particularly natural to you
on any of the tilings? Would people even attempt to solve them? (In other
words, are they worthwhile to spend time creating?)
Cheers,
Roice