Words have weights in a context or category. Change the context to change the weights. Heavy may become light, and lightweight heavy. Weight and momentum go together. A heavyweight word has more inertia meaning impacting its trajectory takes more work.
Category Theory has penetrated higher mathematics by osmosis, as a shared glue language. The objects (nouns) and morphisms (verbs) that make up a category do not require much infrastructure beyond transitivity and associativity. The vocabulary from sets and functions flows in to provide bulk (grist for the mill) right from the get go.
How will Synergetics, over time, affect the weights in CT (or in LLMs more generally)? Polyhedrons are the obvious 4D objects, but we need not include "all" the polyhedrons. We're happy with a subset and Occam's Razor, and the adage: don't add what you don't yet need (or "only add on demand" in other words). Necessity is the mother of invention. Superfluous invention is a mother of clutter and excess.
In the Synergetics namespace, we may start with the Platonics and their duals, the tetrahedron being self-dual. By "dual" we mean vertices exchanged with faces, the number of edges holding constant.
Then we "marry" (Platonic, Dual) pairs to "beget" the rhombics:
- (Tetrahedron, Tetrahedron) begets a Cube (a Rhombic Hexahedron)
- (Cube, Octahedron) begets a Rhombic Dodecahedron
- (Icosahedron, Pentagonal Dodecahedron) begets a Rhombic Triacontahedron
Euler's V+F == E+2 gets introduced, along with Descartes' Deficit and concavity/convexity. We're interested in polar pairs, associated spin axes, and great circle networks (especially juxtaposed and reduced to LCD triangles).
We're also interested in dissections, of polyhedrons into component polyhedrons, and a relative volumes hierarchy.
The Jitterbug Transformation has to qualify as one of the hallmark morphisms, or as a sequence, or poset, of morphisms.
An associative sequence of morphisms might be called a "train" with transitivity implying "express trains" that skip stops. The cuboctahedron to icosahedron "local" then continues to the octahedron, whereas an "express" might go from the octahedron back to the cuboctahedron without click-stopping at the icosahedron.
The connected volumes of 20, 18.51..., 4 count as properties of the polyhedrons in question. Objects have properties.
The cuboctahedron of 2.5 grows to an icosahedron of 2.91796... with two applications of the S:E scale factor (S-factor) where S, E are specific polyhedra (see BEAST modules).
One application of the S-factor would take us to a local station stop of intermediate volume 2.5 < v < ~2.918 tetravolumes. We locate the 12 vertices along the "rails" of a contextualizing octahedron of volume 4.
A question arises as to whether IVM-space and XYZ-space should be considered two different categories, given they contain identical objects and morphisms.
We're saying they could be.
Their isomorphism is clearly apparent, but for the difference in the volume property, which we can iron out. The polyhedron volumes differ by a multiplicative constant.
This Synergetics Constant (S3) suggests itself as a functor in case we do want to separate these 4D and 3D spaces. A cube of edges √2 has volume 3 in IVM world, given its R-edged cube of 1.06066... where R is the radius of any IVM ball (IVM = isotropic vector matrix = the CCP lattice when it comes to balls).
The modules themselves morph into one another, as when the A module morphs into a B module of equivalent volume. Then of course we have φ scaling e.g ...s3, S, S3... and so on, where the S volume is (1/2)(1/φ)5. See my Replit on the S&E modules.
