Adaptive Simplicial Manifold Explorer
A mesh that tests its own wiring.
A mesh is a network of connected triangular regions. Adaptive surgery tests whether local connections can be rearranged to improve a chosen computational objective while preserving specified structural constraints.
Click a vertex to inspect its adjacency.
σ = 10 · ρ = 28 · β = 8/3 · Δt = 0.005 · ε = 0.12 (documented baseline). Changing size, seed or constraints resets both engines.
Diagnostics · adaptive mesh
Objective J over time
One seed, one run — a demonstration, not evidence. Use Experiments for multi-seed comparisons.
Face-quality distribution
q = 4√3·A / (ℓ₁² + ℓ₂² + ℓ₃²); 1 = equilateral, 0 = degenerate.
Vertex degree distribution
Flips move connections around but conserve the total: Σ degree = 2E.
Surgery decision log
Every candidate, its constraint checks, and the measured ΔJ. Accepted only if valid and ΔJ < 0 (best candidate per round).
Step through recorded states
A snapshot of the adaptive engine is recorded at every surgery round. Scrubbing pauses the simulation.
Model definitions (browser reference)
E_field = mean over edges of ‖sᵢ − sⱼ‖² ÷ (2·σ²_ref), where σ²_ref is the total per-coordinate variance of a single Lorenz trajectory at the current parameters (dimensionless; ≈ 1 for unrelated neighbours, 0 for perfect agreement).
E_geometry = mean over faces of (1 − q).
J = E_field + 0.30 × E_geometry. Sync error = √(mean edge ‖sᵢ − sⱼ‖²).
The repository's exact normalisations may differ; these are stated so that every number on this page can be checked.
The edge flip
Two triangles, one diagonal, one decision.
Two adjacent triangles share one diagonal. Surgery removes that shared edge and introduces the alternative diagonal across the same quadrilateral. Vertex, edge and face counts are unchanged, so the Euler characteristic is preserved — but which vertices are neighbours changes, and so do the triangle shapes.
A flip is valid only if the edge is interior, both faces are consistently oriented, the quadrilateral is convex (otherwise a triangle would invert), the new diagonal doesn't already exist, and both new faces meet the quality floor. Validity is necessary, not sufficient: it must also lower J.