some topology

In the yearly tradition of rereading Wolfram's original “Theory Of Everything” and seeing if anything had come of it yet, I realised 2 profound things:

  1. Nothing has come from Wolfram's theory of everything.
  2. Except pretty graphs!

So let's render some.

Each shape emerges from recursively applying a rewrite rule to a set :)

Contact sheet of rendered topology and hypergraph objects

iterated systems

Koch substitution
Heighway dragon
Sierpiński substitution
Recursive cubic hypergraph dust
Menger wire hypergraph
Tetrahedral substitution
Four-dimensional simplex substitution

rewrite dynamics

Wolfram paper headline rewrite rule
Wolfram paper emergent planar rewrite
Wolfram paper emergent three-dimensional surface
Wolfram paper ternary self-loop rewrite
Ternary self-loop subdivision
Wolfram doubled folded fan rewrite rule
Wolfram bidirectional star rewrite rule
Wolfram persistent path closure rewrite rule

finite designs

Fano plane PG(2,2)
Projective plane PG(2,4)
Projective plane PG(2,31)
Affine plane AG(2,32)
Cyclic Steiner triple system STS(13)
Paley 2-(11,5,2) difference design
Complete 3-uniform hypergraph K(10,3)

coset geometry

Cyclic subgroup cosets Z6 modulo the subgroup generated by 2
Square-free cyclic coset geometry Z30030

sporadic objects

Extended Golay code weight shells
Derived Witt S(3,6,22) hexads
Higman-Sims strongly regular graph

simplicial topology

Reversible two-dimensional Pachner edge flip
Periodic simplicial torus
Boundary of the 4-simplex
Boundary of the 4D cross-polytope