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.