Message #3499

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Earthquake Puzzle
Date: Tue, 16 Aug 2016 22:34:13 -0500

>
>
> 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.

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/~cheritat/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.

> Great stuff, Roice!
> -Melinda
>
>
Thanks! Great thoughts from you as well. This is fun!

Best,
Roice