# 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