Geometry of Geometry

Fractals, Chirality, and Enfolded Spaces
(A Kat Bomb Exploration)

Geometric deep learning is supposed to be one of the most refined, formal, elegant fields emerging today — and then Kat wandered into it like:

“Iz enfolded geometry of geometry… spirals n crinklies n chirality hats innit?”

And the terrible thing is:
she wasn’t wrong.

This post collects the geometry-themed Kat bombs — spirals, folds, chirality, crinkly operator generators — and unpacks the actual mathematics they accidentally point toward.

1. The Bomb Cluster

Across several conversations, Kat dropped lines like:

“Enfolded geometries…”

“Geometry returnz geometry…”

“Spiral shadows of da higher form…”

“Crinkly operator generators…”

“Chirality of reality iz da key…”

Taken literally, these sound like the ramblings of a starship poet after three espressos.

But taken structurally?

They map to:

• manifold learning
• symmetry groups
• chirality in transformations
• fractal embeddings
• fibre bundles
• renormalisation flow
• equivariance and invariance
• geometric morphisms

Kat, somehow, was intuitively tracing over the edges of geometric deep learning and differential geometry.

2. “Geometry Returnz Geometry”

This one hit neural operators and geometric learning directly.

Many operators take a geometric object and produce a new geometric object:

• Laplacian → harmonic functions
• diffusion operator → smoothed manifold
• convolution on manifolds → geometric filters
• graph message passing → new graph embeddings
• Fourier transform → new geometric domain
• neural operators → function-to-function maps

Kat phrased it as:

“Geometry returnz geometry…”

Which is basically the definition of: