# Message #3500

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Earthquake Puzzle

Date: Tue, 16 Aug 2016 22:49:47 -0700

On 8/16/2016 8:34 PM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:

>

>

>>

>> In fact, you can do an earthquake twist where all 3 systoles

>> break from the surface instead of just two of them, which makes

>> it a little easier to see why earthquakes are vertex-centered.

>> This is like a 3-cycle rotation about a hub. I didn’t include

>> that twisting because I thought it might make the puzzle easier

>> if more permutation options were allowed.

>

> It’s definitely trippy to see one systole twisting in place while

> the others detach and reattach but I don’t understand how that

> makes anything more clear. The pure vertex earthquake twist is

> more symmetric and I’d expect it would be easier to understand,

> no? Maybe you’re even making it more difficult on yourself to

> support those non-detaching earthquake twists. Was that a big part

> of what made the implementation difficult? In the current case of

> one rotating systole and the rest moving, I’m often reminded of

> the Grand Staircase <https://www.youtube.com/watch?v=uFvizAQHJz8>

> in the Harry Potter movies in which the ends of particular

> staircases detach and reattach to different landings.

>

>

> oh yeah, I agree. The pure vertex earthquake would be easier to

> understand, and would be more easily interpreted as vertex-centered.

> I think my wording was confusing.

>

> When Arnaud and I were talking about this, we were really hoping for a

> twist that slid along the 3 systoles and didn’t detach anywhere. I’d

> like to be wrong, but that seems to be impossible.

Except for the simple genus 1 (torus) case, I’m not certain but I think

you’re right. For some surfaces I think you can slide two of the

systoles but no more.

>

> The main difficulty in implementation was performance. These twists

> affect a large number of tiles and stickers at once on the universal

> cover, which is where the engine is doing it’s internal drawing. I

> had to parallelize some parts of puzzle building to deal with the

> fallout of that, and had to profile and make various optimizations to

> get it to run reasonably well. In fact, there are some rendering

> artifacts I haven’t eradicated. It would be easy to switch to the

> symmetric vertex-centered earthquake now, but I probably would have

> had to deal with the same sorts of performance issues if I had started

> there. The v2 engine is so much better than my first MagicTile

> attempt, but this puzzle pushed it’s boundaries. At this point, I

> think I have a sense of how a better v3 engine could be designed, but

> I doubt I’ll ever do it.

>

> The comparison to the grand staircase does seem very apt!

>

> Now here’s a truly crazy idea. In trying to imagine both the 2D

> and 3D aspects together, I imagined the current 2D view as a plane

> in 3-space, intersected by 3D arches. One could initiate

> earthquake twists on the 3D structure, and 2D twists in the plane.

> Looked at this way it bares a striking resemblance to that amazing

> rendering you did called Hyperbolic Catacombs

> <http://gallery.bridgesmathart.org/exhibitions/2015-joint-mathematics-meetings/roice3>.

> I never did follow what that thing was so I have no idea if shares

> any connection with this puzzle, but if it does, then it suggests

> the possibility of a wonderfully immersive VR puzzle.

>

> This got me thinking of MC2D and how the reflection twists are like 3D

> rotation twists out of the plane. Perhaps these earthquake twists are

> also "higher-dimensional" in this sense, and maybe there is some

> natural out-of-plane representation like you are describing which

> would be cool.

>

>> I think the twist you described earlier would be an edge-centered

>> earthquake.

>

> Exactly. The only missing analog would be an earthquake face

> twist. On this puzzle, that’s equivalent to a twist of the

> opposite vertex but in larger puzzles may be interesting though

> the UI challenges makes me think it’s probably not worth attempting.

>

>

> I hadn’t thought through the face-centered earthquake case yet, but

> using Arnaud’s applet

> <http://www.math.univ-toulouse.fr/%7Echeritat/AppletsDivers/Klein/> I

> just convinced myself there is no possible earthquake face-twist, at

> least not based on systoles. Click the "systolic pants decomp" option

> there and look at, say, the white "pair of pants" in the center. A

> twist will move material within that pair of pants, but at the end of

> the twist the new location of all the shuffled material will need to

> cover the same original area. If you pan certain heptagon vertices or

> edges to the center of the view, you can see that this works. If you

> move a heptagon center to the center of the view, it doesn’t - there’s

> no way to make a 1/7th turn and get the pants to return to covering

> the original area.

OK, well now I’m having an out-of-pants experience but it made me

realize that I didn’t previously make myself clear. Once I clarify

myself, you may conclude that I indeed have gone off into the weeds

regarding catacombs. When talking about the "meta" puzzle, I was

referring to the topology of the surface itself as opposed to the puzzle

within it though maybe they’re always identical. In the case of KQ, the

genus is 3 making the topology that of a ball-and-stick model of a

tetrahedron. When I talk about vertex twisting at the meta level I’m

talking about cuts through the arms of this tetrahedron

<http://math.ucr.edu/home/baez/pentacontihexahedron2.jpg>. So your

current twists appear to cut three arms, twist one of them in place by

180 degrees while swapping the other two. What I called the pure

"vertex" twist would sever three arms that meet at a meta-vertex, rotate

that whole unit 120 degrees and reattach them all. The "edge" twist cuts

4 arms straight through the center of the tetrahedron and rotates one

half by 180 degrees. By now you probably understand what I mean about

"face" twists, which in the case of KQ is identical to a pure vertex

twist opposite a given systolic triangle.

-Melinda