Topological face extraction and Euler target rings

To turn an elastic bouncing buckyball into an arcade physics challenge, the engine needed to recognize molecular faces rather than just raw vertices and edges. Implemented a chordless minimal cycle traversal algorithm that parses the raw node-link graph to automatically detect 2-manifold faces, extracting exactly the 12 pentagons and 20 hexagons governed by Euler's polyhedral formula ($V - E + F = 2$). Transformed the 12 pentagonal topological defects into glowing holographic target rings oriented along their outward surface normals. Built a high-speed projectile slingshot mechanic that lets you fire particle probes through the molecular cage, rewarding precision ring threading with procedural pentatonic chime sweeps and exponential combo multipliers when ricocheting off interior nodes.