Thread: "Earthquake Puzzle"

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Wow!

Roice and Melinda your are amazing!

Kind regards
Ed

----- Original Message -----=20
From: Melinda Green melinda@superliminal.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Monday, August 15, 2016 3:42 AM
Subject: Re: [MC4D] Earthquake Puzzle


=20=20=20=20
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 twist=
s.

You've probably already guessed what I'm going to which is whether this p=
uzzle 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 underst=
anding the topology of the puzzle. I stared a bit at this image to try to u=
nderstand 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 c=
orrect? That seems to suggest that the scrambling twists plus solving twist=
s 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. Actuall=
y, it looks like there are more than one way to do that too though the simp=
le geometric rotation seems the most natural.

The really neat thing about your new feature is that it works at a kind o=
f meta level by operating on the hubs and struts of high genus surfaces sim=
ilarly to how we've been twisting vertices and edges within them. Heck, it =
looks like you could even create puzzles within puzzles where you manipulat=
e the structure like you are doing now while also allowing users to twist t=
he 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 th=
e {7,3} or other IRPs supported in this way. None of this is to pressure yo=
u to implement anything but rather to try to understand what this new puzzl=
e 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,=20


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 p=
oint, we slice it up with systolic (shortest length) geodesics. These geod=
esics cut the surface "around the horn" as Melinda has described in the pas=
t. 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 th=
e 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 t=
emporarily detach the surface from itself during the course of a twist. Th=
e surface remains connected along one systole (the movement near this slice=
reminds me of the "Big Chop" puzzle), but detaches along the other two sys=
toles, swapping the material connected to each of them. The twist animatio=
n hopefully gives a flavor of the surface separating and reattaching to its=
elf.=20=20


I attempted to make it intuitive to control an earthquake twist, but no=
te there are three ways to twist a set of systoles (6 if you count directio=
n, 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 puzz=
le pristine and scrambled.


I have not tried to solve this puzzle yet. I hope it is a good challen=
ge, 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 im=
plement!=20=20


Grab the latest MagicTile, check it out, and give us some detailed note=
s of anything you discover while solving it!


Cheers,
Roice











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Thanks for your response Melinda. Your email really adds to the
description of the twisting! I have some inlines below...

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?
>

Yes, exactly!


> 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.
>

Very cool, I hadn't considered this. I think the twist you envision will
swap two sets of systoles, with four cuts total, and all of them will
detach during the twist.

Your idea made me think of something else as well... I had thought about
systolic twists on a torus puzzles, which could be done without detaching
the surface at all. But I hadn't thought about doing twists with two
"around the horn" cuts on a single strut of the genus-3 surface. I'm
guessing the cuts wouldn't be geodesics or shortest length in this case,
but it still does seem like it should be possible. I'll have to think on
that more.

Lots of possibilities!


> 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?
>

Yeah! One thing about the earthquake twists is that they are "centered" on
vertices, so a natural "twist within a twist" would be the normal
vertex-centered twist MagicTile already supports.

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. Plus the controls
were already difficult to try to make intuitive.

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


> 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.
>

Totally, I want to see this too, and I have some surprises in this
direction I've been working on with Burkard. I'm nowhere near there yet
with the earthquake puzzle, and not even there with the classic KQ, but we
will have some cool stuff to show. Hopefully what you are picturing will
come eventually too!

Best,
Roice

P.S. I've been checking all changes into GitHub, and anyone who is
interested in participating in the development is welcome. Hopefully
looking at recent code diffs could help others find their way around the
code base.

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