my work, AI-generated overview

Ruslan Mizhaev’s Integer Realization of an Equivelar Octahedron of Genus 3 does something I like: it takes an object previously known only abstractly and pins it down with exact integer coordinates, so every claim about it is checkable without a single floating-point comparison. Eight planar nonagonal faces, 24 vertices, 36 edges, three faces meeting at every vertex — a {9,3} equivelar map — with C₄ symmetry. Twenty pairs of faces share an edge; the remaining eight pairs share two.

The genus follows from counting: 24 − 36 + 8 = −4, and χ = 2 − 2g gives g = 3.

But counting is not seeing. The faces are big thin nonagons that weave past one another, and no still picture made it obvious to me that this spiky thing is a triple torus. So I built an animation that relaxes the polyhedron into a smooth genus-3 surface under surface tension at constant volume. Every keyframe, and every linear interpolation between keyframes, is certified free of self-intersection, so the surface never cheats by passing through itself. Each triangle keeps the color of the flat face it started on. By the end the three tunnels are hard to miss.

Drag to rotate, scroll to zoom, and use the opacity slider to see through the surface. The F1–F8 checkboxes show and hide individual faces: switch off all but two and you can see exactly where those two meet, for any pair you care to pick — easiest a little way into the morph, once the razor-thin nonagons have fattened into ribbons. Every one of the 28 pairs touches, which is the 20 + 8 above: each face meets every other, so the map is neighborly. It loads about 4 MB on demand, so nothing downloads until you press the button. If the embed is too cramped, open it on its own page; the viewer and the animation data are on GitHub at murbard/genus3-morph.