# 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

1. **Height field** — for each (x, c) grid point, compute d = (−6x⁵ + 12x³ − 2cx) × D_SCALE
2. **Mesh construction** — adjacent grid quads are triangulated into a regular height-field mesh
3. **Base slab** — a flat rectangular base is added so the model is self-supporting on a print bed
4. **ASCII STL output** — written as ASCII (not binary) for maximum compatibility with slicers and viewers

---

## Running the Generator

```bash
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).