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Chainmail Homopolymers in $\mathbb{R}^3$
this will be converted into a table of /basic/ (which is a term I'll have to find a definition of later)
chainmails of platonic polyhedral skeleta
paulbourke.net/geometry/chainmail/
definitions:
- lattice-structure torsor a lattice-structure torsor is sort of what an individual point/section of a lattice structure
sees. To be physically realizable: there must be some additional structure, or in the infinite, how do
we know that we're not a solipsist point/section that, to pass the time, has pretended to be two different
points/sections. The practical consequence of lattice torsors is losing one's place in a repeating structure
- plenum space (this will eventually be important to Dehn-theoretic thinking)
- sashing space/distance
Martha Ingols:
- splunge space/distance
Scott Lamb
the sashing space realized in the lattice structure: the "sashing plunged through other sashing, is splunge"
- Coordination number
has the same meaning as in /Sphere Packings, Lattices, and Groups/
because a subset of the corners of a dodecahedron correspond to a cube, one can generally
take the chainmail produced from cubes and replace them with dodecahedra and one will generally
produce another. the following table only covers some pretty simple chainmail homopolymers of
tetrahedra, hexahedra, and dodecahedral skeleta.
Realization and Riemann Surfaces
there are Riemann surfaces which can be produced from these skeleta by assuming that each
vertex is replaced by a pair of pants. For characterizing (without respect to particular
kinematics or dynamics) the vibrational modes of these lattices, I'm going to take the somewhat
unusual step of thinking of each monomer as a Riemann surface, because for 'can haz here's a tetrahedron'
in R^n, We don't really have a good model of 'here's mathematical interior, not a physical object', I don't
have a good feeling I'm dealing with a whole object. Whereas (with these chainmails): if you just have
vertices and edges and no other structure, how vibrational modes will propagate in monomers is
difficult to get at
atoms don't work like these chainmail homopolymers
here are a few papers which are about Platonic Riemann surfaces:
https://link.springer.com/article/10.1007%2Fs000140050082
- the rings chainmail decomposes into three repositioned rings in standard square lattices with sashing ---
- the dodecahedral edge-plenum linked chainmail homopolymer decomposes into two 3d checkerboard lattices (each coordinates with six others, it is not saturated)
- a -one and its nearest neighbors- are 27, form a cube at rest
- when hanging from a corner, forms an acute golden rhomboid
- when squashed on a corner-to-corner diagonal, forms an obtuse golden rhomboid
- has a (comfortable) completely-touching/completely compressed format
- is (uncomfortable) at tension because there are just edges touching edges
- your impression is /because they touch at a point, they can slip)
- the vertex-plenum linked tetrahedrral chainmail homopolymer decomposes into two hcp with this sashing...
(each coordinates with four others. it is saturated)
- the translation invariant vertex-plenum linked dodecahedral chainmail homopolymer does not decompose
- block copolymers decompose with respect to Q^3 this way...
- is comfortable at tension (corners fit in corners), is not comfortable at compression
- there's twisted vertex plenum linked chainmail homopolymer of cubes
- there may be a homopolymer with twisted rings of cubes
- the half density translation invariant vertex plenum linked dodecahedral chainmail homopolymer
- edge plenum linked hexahedral chainmail homopolymer
- vertex plenum linked hexahedral chainmail homopolymer
- half-density vertex plenum linked chainmail homopolymer
I should deconstruct a little what the motivation for trying to organize this is:
Octobers are usually dedicated to lattices, and as per
Physicists Aim to Classify All Possible Phases of Matter, it does make sense to have a rudimentary taxonomy of three
dimensional chainmails. I don't know how these ramify with respect to modular tensor categories. There are some other
lattice structures in three dimensional space I want to spend more time on, and arguably, making pictures of them
might be a productive thing to do, but at the moment, I don't really have much in the way of either institutional
or social support for this endeavour, so I trundle on.
Bounty: I am offering a bottle of whisky to anyone who can successfully
translate Jacob Lurie's
Higher Topos Theory (and/or
the stacks project) into
the
analytic combinatorical language of the manner of Flajolet's
Analytic Combinatorics.