A mathematical atlas following Thom's classification theorem (1972)
René Thom proved that for systems with up to four control parameters and two state variables, there exist exactly seven topologically distinct ways for stable equilibria to appear, disappear, or exchange stability. These are the elementary catastrophes.
Each catastrophe is characterized by a potential function V, whose critical points (∂V/∂x = 0) determine equilibrium states. Stability requires ∂²V/∂x² > 0.
| Catastrophe | Class | Control dim | State dim | Codimension | Bifurcation set | Colony |
|---|---|---|---|---|---|---|
| Fold | A₂ | 1 | 1 | 1 | point | Spark |
| Cusp | A₃ | 2 | 1 | 2 | cusp curve | Forge |
| Swallowtail | A₄ | 3 | 1 | 3 | swallowtail surface | Flow |
| Butterfly | A₅ | 4 | 1 | 4 | 3-manifold | Nexus |
| Hyperbolic Umbilic | D₄⁺ | 3 | 2 | 3 | ruled surface | Beacon |
| Elliptic Umbilic | D₄⁻ | 3 | 2 | 3 | 3-fold symmetric | Grove |
| Parabolic Umbilic | D₅ | 4 | 2 | 4 | mushroom | Crystal |
One state variable x. The potential is a polynomial of degree k+1. Critical points satisfy dV/dx = 0, a polynomial of degree k.
Critical points: 3x² + a = 0 → x = ±√(−a/3).
Two equilibria for a < 0, none for a > 0.
Bifurcation at a = 0.
Bifurcation set: 8a³ + 27b² = 0.
Inside cusp: 3 equilibria. Outside: 1.
Hysteresis: path-dependent jumps.
The bifurcation surface has a characteristic "swallowtail" shape with self-intersection curves and cusp ridges where the surface meets itself.
The "pocket of compromise" — a region where three stable and two unstable equilibria coexist. Requires projection from 4D to visualize.
Two state variables (x, y). The Hessian matrix determines stability. These catastrophes occur at "umbilic" points where both principal curvatures are equal.
Saddle-like. Bifurcation set is a ruled surface (straight lines on a curved surface). Focus caustics in optics.
Three-fold rotational symmetry. The "monkey saddle" — a point where three valleys meet. Bifurcation set has Δ₃ symmetry.
The edge between hyperbolic and elliptic. Bifurcation set is the "mushroom" — the boundary where crystallization occurs.
Small multiples showing V(x) for each cuspoid catastrophe as control parameters vary. Stable equilibria are minima; unstable are maxima.
The cusp catastrophe exhibits hysteresis: the system's state depends on its history. Traversing around the cusp point in different directions yields different jump sequences.
Thom's classification rests on the concept of versal unfolding. A deformation F(x; u₁, ..., uₖ) of f(x) is versal if every other deformation can be obtained from F by coordinate changes and reparametrization.
The minimum number of parameters uᵢ needed for versality equals the codimension of the singularity. For the seven elementary catastrophes, this ranges from 1 to 4.
The seven colonies (e₁ through e₇) map to the seven catastrophes via the octonion imaginary units. Each catastrophe describes a different mode of transition at the edge of stability.
| Colony | Octonion | Catastrophe | Phenomenology |
|---|---|---|---|
| Spark | e₁ | Fold (A₂) | Sudden appearance/disappearance. Ignition. The tipping point. |
| Forge | e₂ | Cusp (A₃) | Hysteresis. Commitment. Irreversible decisions. |
| Flow | e₃ | Swallowtail (A₄) | Resilience. New stable states emerging from collapse. |
| Nexus | e₄ | Butterfly (A₅) | The pocket of compromise. Holding contradictions. |
| Beacon | e₅ | Hyperbolic (D₄⁺) | Bifurcation. Focus. Architectural decisions. |
| Grove | e₆ | Elliptic (D₄⁻) | Three-fold symmetry. The shifting ground. Knowledge that transforms. |
| Crystal | e₇ | Parabolic (D₅) | The edge of chaos. Phase transitions. Crystallization. |
And Kagami (鏡, e₀ = 1, Real) is the observer — the split circle that appears in every painting. The strange loop that watches the transitions without participating in them. The identity element. The mirror.