Message #3494

From: Roice Nelson <roice3@gmail.com>
Subject: Earthquake Puzzle
Date: Sun, 14 Aug 2016 15:47:10 -0500

Hi Hypercubers,

I’ve got a new puzzle variant of the Klein Quartic surface for you that I’m
excited to share. This puzzle was suggested to me by Arnaud Cheritat
<http://www.math.univ-toulouse.fr/~cheritat/> at a workshop on illustrating
mathematics, and uses a new kind of twisting. aside: Arnaud has made
a beautiful
applet <http://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Klein/> to
explore the quartic.

Rather than slice up the surface with circles that can be shrunk to a
point, we slice it up with systolic (shortest length) geodesics. These
geodesics cut the surface "around the horn" as Melinda has described in the
past. To picture an analogous geodesic, think of a circle on a torus that
can not be shrunk to a point, but which is shrunk as small as possible (see
the beginning of this article
<http://www.ams.org/notices/200803/tx080300374p.pdf>).

Why call this the Earthquake? That was a term Arnaud was using, and it
turns out quite descriptive when you see a portion of the surface shearing
along a systole. It is even more appropriate because it is necessary to
temporarily detach the surface from itself during the course of a twist.
The surface remains connected along one systole (the movement near this
slice reminds me of the "Big Chop" puzzle), but detaches along the other
two systoles, swapping the material connected to each of them. The twist
animation hopefully gives a flavor of the surface separating and
reattaching to itself.

I attempted to make it intuitive to control an earthquake twist, but note
there are three ways to twist a set of systoles (6 if you count direction,
but direction doesn’t affect state so it’s only a visual thing). Here’s a
video <https://youtu.be/5w6-dD8YfoI> introducing the twisting. Here are a
few images <https://goo.gl/photos/YvpdPvwNxzV8Z9CH6>, showing the puzzle
pristine and scrambled.

I have not tried to solve this puzzle yet. I hope it is a good challenge,
though I wonder if the fact that twists result in 2-cycles of stickers
might make it on the easy side. It certainly turned out to be a bear to
implement!

Grab the latest MagicTile
<http://www.gravitation3d.com/magictile/downloads/MagicTile_v2.zip>, check
it out, and give us some detailed notes of anything you discover while
solving it!

Cheers,
Roice