# Message #3496

From: Eduard Baumann <ed.baumann@bluewin.ch>

Subject: Re: [MC4D] Earthquake Puzzle

Date: Mon, 15 Aug 2016 09:40:45 +0200

Wow!

Roice and Melinda your are amazing!

Kind regards

Ed

—– Original Message —–

From: Melinda Green melinda@superliminal.com [4D_Cubing]

To: 4D_Cubing@yahoogroups.com

Sent: Monday, August 15, 2016 3:42 AM

Subject: Re: [MC4D] Earthquake Puzzle

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 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 at a workshop on illustrating mathematics, and uses a new kind of twisting. aside: Arnaud has made a beautiful applet 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).

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 introducing the twisting. Here are a few images, 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, check it out, and give us some detailed notes of anything you discover while solving it!

Cheers,

Roice