# Message #3495

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Earthquake Puzzle

Date: Sun, 14 Aug 2016 18:42:00 -0700

How very cool, Roice!

I don’t quite understand how the twist highlighting works but I’m able

to sort of find the ones I want and solve a couple of random scrambling

twists.

You’ve probably already guessed what I’m going to which is whether this

puzzle could support the "Show as Skew" view. We’ve learned that solving

is better supported by the hyperbolic view but a 3D view would help in

understanding the topology of the puzzle. I stared a bit at this image

<http://math.ucr.edu/home/baez/pentacontihexahedron2.jpg> to try to

understand what’s going on. From your description it sounds like each

twist cuts one off the three struts of a red hub, turns one of those

arms around its cut, flipping the hub over and swapping the other two

struts. Is that correct? That seems to suggest that the scrambling

twists plus solving twists is always even. It also suggests there are

other types of possible twists. One of them seems like the most natural

one to me which twists a selected strut by 180 degrees, swapping the

hubs at each end. That one seems to be a "true" Big Chop-like deep cut

since it’s symmetric on both sides. Actually, it looks like there are

more than one way to do that too though the simple geometric rotation

seems the most natural.

The really neat thing about your new feature is that it works at a kind

of meta level by operating on the hubs and struts of high genus surfaces

similarly to how we’ve been twisting vertices and edges within them.

Heck, it looks like you could even create puzzles within puzzles where

you manipulate the structure like you are doing now while also allowing

users to twist the elements within the texture with a modifier key or

something. Does that make any sense?

Assuming I haven’t gone completely off into the weeds, I’d love to see

the {7,3} or other IRPs supported in this way. None of this is to

pressure you to implement anything but rather to try to understand what

this new puzzle means and where it could go.

Thanks for the wonderful new toy!

-Melinda

On 8/14/2016 1:47 PM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:

>

>

> 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/%7Echeritat/> 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/%7Echeritat/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

>

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