Topology, geometry & mathematical invariants
What changes, what can't, and what we measure.
Adaptive surgery is only meaningful if we are precise about three different kinds of quantity: topological invariants that valid flips must preserve, geometric quality that flips are allowed to change, and performance metrics that only experiments can judge.
The documented objective
Efield — mismatch across the modeled system
Measures how different neighbouring vertex states are across mesh edges. Because edges define who is a neighbour, rewiring changes which differences are counted — and, through coupling, how states evolve afterwards. In the browser reference: mean edge ‖sᵢ − sⱼ‖² ÷ 2σ²_ref.
Egeometry — penalty for poor triangles
Penalises badly shaped faces. In the browser reference: mean of (1 − q) with q = 4√3·A / Σℓ². The weight 0.30 sets the trade-off: a flip may worsen geometry slightly if it improves field agreement enough, and vice versa.
Exact definitions are those implemented in the source repository; the labs state their own so every number can be checked. Lower J under the same constraints is the acceptance criterion — it is not, by itself, evidence of better computation.
Before and after
An accepted local transformation, measured.
This 6 × 6 mesh uses the geometry term only (no dynamics), so you can isolate what a flip does. Compare an edge flip — topology preserved — with deleting a face, which genuinely changes topology.
BEFORE
AFTER
| Quantity | Kind | Before | After |
|---|
Green = unchanged, amber = changed. Removing a face keeps its three edges as 1-simplices, so χ drops by one and a 1-cycle (hole) appears: b₁ = b₀ − χ + b₂.
Persistent topology
Which features survive across scales?
Grow a radius ε around each point and connect points closer than ε. Components merge as ε grows; the H₀ barcode records when each component dies. Long bars are robust structure; short bars are noise.
H₀ barcode
Computed exactly with a union–find (single-linkage) filtration in your browser. The project's persistence-related diagnostics may use different filtrations; this panel explains the idea, it does not reproduce repository output.
Three kinds of quantity — never interchangeable
| Kind | Examples | Under a valid edge flip | What it can tell you |
|---|---|---|---|
| Topological invariant | χ, b₀, b₁, orientability, boundary components | Preserved by construction | The flip didn't tear, glue or invert the surface. Says nothing about performance. |
| Geometric quality | Triangle quality q, min angle, distortion, degree distribution | Changes | Mesh shape and numerical conditioning. |
| Performance metric | J, synchronization error, runtime, benchmark scores | Changes over time | Whether adaptation helps — only via matched, repeated experiments against fair baselines. |