research/origami/physics.md

Physics of Paper Folding

Fundamental Theorems

Kawasaki's Theorem (Angles)

• At a single vertex, flat-foldable iff alternating angle sums = 180 on each side
• Necessary but not sufficient for global flat-foldability
• Use as: reward penalty for violation

Maekawa's Theorem (Mountain/Valley Count)

• At any flat-fold vertex: |M - V| = 2
• Total creases at a vertex must be even
• Use as: hard constraint in action validation

Global Flat-Foldability

• NP-complete (Bern & Hayes, 1996)
• Hardness comes from determining valid layer ordering
• Local conditions (Kawasaki + Maekawa) necessary but not sufficient
• Use as: approximate via constraint satisfaction for reward

Rigid Foldability

• Can it fold from flat to folded state with rigid panels (no face bending)?
• Critical for engineering: metal, plastic panels must be rigid
• Checked via: triangle-triangle intersection + rigid body constraints
• Use as: pass/fail validation check

Layer Ordering

• When paper folds flat, layers stack — which face is above which?
• Must satisfy: no face penetration
• FOLD format: faceOrders triples [f, g, s]
• This is the NP-hard part — approximate for RL

Simulation Approaches (Ranked for RL Use)

1. Bar-and-hinge (Ghassaei) — edges as bars, rotational springs at hinges. Fast. Best for RL.
2. Rigid body + compliant creases — rigid panels, torsional spring creases. Good middle ground.
3. FEM — full stress/strain tensor. Accurate but too slow for RL training loop.