# Message #3605

From: Roice Nelson <roice3@gmail.com>

Subject: 1000+ MagicTile Puzzles

Date: Tue, 03 Jan 2017 22:45:56 -0600

I just uploaded the latest MagicTile, which includes a number of new

colorings and slicings, bringing the total puzzle count to 1020. I haven’t

updated the wiki with slots for recent additions, but I did generate full

puzzle lists

<https://beta.groups.yahoo.com/neo/groups/4D_Cubing/files/MagicTile/> before

and after this change, so you can diff those two files to see what was

added.

Some puzzles are using new code that allows defining colorings via group

relations of a regular map. I’ve appended the names of these with their

designation from this page

<https://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt>.

I’d like to point out one of the new regular maps and relate it to

something interesting about hyperbolic 2-manifolds.

The dual {4,7} 42-color and {7,4} 24-color share the same symmetry group as

the Klein quartic (168 orientation-preserving symmetries), but these

surfaces are *genus 10* rather than genus 3. The Gauss-Bonnet theorem

tells us The Gauss-Bonnet theorem tells us the area of a hyperbolic

2-manifold is a function of its Euler characteristic, χ

<http://claymath.msri.org/gabai.m4v>

<http://claymath.msri.org/gabai.m4v>

A = -2*π*χ

So high genus surfaces (with more negative Euler characteristic) have

larger areas.

I wonder if solving a simple slicing of the latter (F0:0:1 or E1:0:0, say)

would feel different than the same slicing of the KQ. My guess is that

even though it has the same number of colors and a larger area, it may

somehow feel more cramped (in a similar way to how the Rubik’s cube feels

cramped compared to Megaminx). I’ll have to try.

A quick aside: I like the organization of the spherical and elliptical

puzzles in the MagicTile tree, but the hyperbolic folder feels like a

mess. I’m thinking about organizing by genus, then maybe

orientable/non-orientable/orbifold under that. This would scatter the same

Schläfli symbols throughout the tree though, so I’m not sure. If folks

have opinions on what would be best, I’d appreciate them.

Happy 2017 everyone!

Roice