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

Butterfly Catastrophe

Potential: V(x) = x⁶ + ax⁴ + cx² + dx (a = −3 fixed, b = 0)
Equilibrium condition: dV/dx = 6x⁵ − 12x³ + 2cx + d = 0
Rearranged as a height field: d = −6x⁵ + 12x³ − 2cx
Print base: (x, c) plane — state variable × control parameter
Print height: d (the other control parameter, computed directly)

This parameterisation is single-valued: every (x, c) point maps to exactly one d, so the mesh is a simple height field with no multi-valued branches, no root finding, and no fold-edge gaps.

The fold ridges — where the surface has zero gradient in x — satisfy ∂d/∂x = 0, giving the bifurcation curve c = 18x² − 15x⁴. This self-intersecting curve is visible as a characteristic ridge on the surface.

Why not sweep (c, d) and solve for x? That approach requires finding multiple roots of a degree-5 polynomial at each grid point, tracking which root belongs to which branch across fold lines, and capping fold edges — all of which introduce artefacts and holes. Rearranging to d(x, c) avoids all of this entirely.

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

Height field — for each (x, c) grid point, compute d = (−6x⁵ + 12x³ − 2cx) × D_SCALE
Mesh construction — adjacent grid quads are triangulated into a regular height-field 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`	`200`	Grid resolution — higher = smoother fold ridges
`X_RANGE`	`(-3.0, 3.0)`	Range of state variable x (print width)
`C_RANGE`	`(-1.0, 5.0)`	Range of control parameter c (print depth)
`D_SCALE`	`0.5`	Vertical scale factor — reduce if the model is too tall

The butterfly fold structure is concentrated around x ∈ [−1.1, 1.1] and c ∈ [0, 5.4]; extending X_RANGE beyond ±2 adds flat outer wings with no additional features.

3D Printing Tips

Orientation: flat base down — no supports needed
Layer height: 0.15–0.20 mm for good surface detail
Perimeters: ≥ 2 for the thin ridge regions
Scale: the fold ridges are most visible at ~100–150 mm along the c-axis
Material: PLA or PETG both work well; the overhangs are gentle

Dependencies

numpy

No other dependencies — STL writing uses plain file I/O (ASCII STL).

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