Message #3505

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Earthquake Puzzle
Date: Mon, 22 Aug 2016 14:32:11 -0500

>
>
> 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.

Cheers,
Roice