# Message #3507

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Earthquake Puzzle

Date: Mon, 22 Aug 2016 17:06:01 -0700

On 8/22/2016 12:32 PM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:

>

>> 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.

>

>

>

> I think we have been picturing things really similarly. I was also

> thinking of how the vertex and edge earthquake twists affected that

> thickened "tetrahedron", just like you. If you map the KQ surface to

> that tetrahedron such that a heptagon vertex maps to a tetrahedron

> vertex, it does work identically. And similarly for mapping a

> heptagon edge to a tetrahedron edge.

>

> I hadn’t understood what you meant by a face being opposite a vertex

> before, but now I get that (on the KQ surface itself, each vertex has

> one opposite vertex and each face has two opposite faces). There is

> (of course) no way to map a heptagon center to a tetrahedron face,

> since the latter is not part of the surface, so the equivalence

> between this topology perspective and the surface perspective breaks down.

>

> Here’s some more intuition as to why the equivalence breaks down in

> the face-centered case. The KQ surface has a lot more symmetries than

> a tetrahedron, but the symmetry group of the tetrahedron is a subgroup

> of the symmetry group of the KQ. A symmetry that rotates 1/7th a turn

> about a heptagon center is one of the KQ symmetries that is not a

> symmetry of the tetrahedral subgroup, whereas the vertex-centered and

> edge-centered twists are symmetries of both. Maybe this is also the

> reason why you can’t have a face-centered earthquake twist (on the

> surface), because the systoles are arranged with tetrahedral

> symmetry. I bet I’m being clear as mud, but hopefully this adds

> something.

We may be talking past each other, but when I talk about "face

centered", I’m not talking about heptagons, but rather the four

triangular faces of the thickened tetrahedron. Since it’s a tetrahedron,

those face twists are equivalent to the symmetric earthquake twist of

the opposite "thickened vertex". Talking about twists on this

tetrahedron (AKA "earthquake" twists) is at the "meta" level that I’ve

mentioned. I’m probably just adding more mud but maybe it will help.

-Melinda