Catastrophe Theory — 3D Print Models

Project Overview

This project generates 3D-printable STL models of surfaces from catastrophe theory — a branch of mathematics studying how small changes in parameters can cause sudden, discontinuous changes in a system's equilibrium state.

Two surfaces are being produced:

Model	Catastrophe Type	Codimension	Potential
Cusp	Cusp catastrophe	2	x⁴ + ax² + bx
Butterfly	Butterfly catastrophe	4	x⁶ + ax⁴ + bx³ + cx² + dx

The Mathematics

Each surface is the equilibrium manifold — the set of all points where the system is in equilibrium. For a potential V(x), equilibria satisfy:

dV/dx = 0

The 3D surface is swept over the control parameter space (a, b), with x (the state variable) as the third axis. Where the surface folds back on itself is the bifurcation set — the region where the system can catastrophically jump between states.

Butterfly Catastrophe

Potential: V(x) = x⁶ + ax⁴ + cx² + dx (a = −3 fixed, b = 0)
Equilibrium condition: dV/dx = 6x⁵ − 12x³ + 2cx + d = 0
Control space: (c, d) swept over a 2D grid
State space: up to 5 real roots x at any given (c, d)
Characteristic feature: nested "butterfly wing" fold structure — a self-intersecting bifurcation curve in the (c, d) plane enclosing a 5-root "pocket" (c∈[0,3], d≈0), surrounded by a 3-root wing region, with a single-root region outside

Why not vary (a, b) with c=d=0? With c=d=0, the equilibrium equation factors as x²(6x³ + 4ax + 3b) = 0 — x=0 is always a double root and the remaining roots come from a cubic, which is structurally identical to the cusp catastrophe. The butterfly structure only appears when d ≠ 0 generically, which requires d (or an equivalent odd perturbation) to be varied as a control parameter.

Files

butterfly_catastrophe.py — generates the butterfly surface STL
butterfly_catastrophe.stl — ready-to-slice output (ASCII STL)
CLAUDE.md — this file

A cusp catastrophe script also exists and was the starting point for this project.

How the Generator Works

Root finding — at each (a, b) grid point, all real roots of dV/dx = 0 are found using Newton-Raphson with dense initial seeding across the x range
Branch tracking — roots are sorted and matched by branch index across adjacent grid cells
Mesh construction — adjacent grid quads on the same branch are triangulated into a surface mesh
Base slab — a flat rectangular base is added so the model is self-supporting on a print bed
ASCII STL output — written as ASCII (not binary) for maximum compatibility with slicers and viewers

Running the Generator

python butterfly_catastrophe.py

Output: butterfly_catastrophe.stl

Tuning Parameters (inside the script)

Parameter	Default	Effect
`GRID`	`50`	Resolution of the (c,d) control grid — increase to 80–100 for final print
`C_RANGE`	`(-2.0, 7.0)`	Range of control parameter c
`D_RANGE`	`(-6.0, 6.0)`	Range of control parameter d
`X_RANGE`	`2.5`	Search window for equilibrium roots
`MAX_EDGE_DZ`	`0.6`	Z-jump threshold for rejecting branch-mismatch triangles at fold edges

For a final high-quality print, increase grid to 80–100. The default of 40 is optimised for STL viewer compatibility.

3D Printing Tips

Orientation: flat base down — no supports needed
Layer height: 0.15–0.20 mm for good surface detail
Perimeters: ≥ 2, as the fold regions are thin
Scale: ~120 mm along the a-axis makes a good desk model
Material: PLA or PETG both work well; the overhangs are gentle

Dependencies

numpy

No other dependencies — STL writing uses Python's built-in struct module (binary) or plain file I/O (ASCII).

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