Wrapped Packages

In the overwhelming book The Symmetries of Things  by John Horton Conway, Heidi Burgiel and Chaim Goodman-Strauss there is an image of an object called Octahedral 3¹². You get an essential part of it by placing eight antiprisms over a triangle (or octahedral annuli) inside eight cubes like so:

The precise placement of the octahedra is so that the yellow dots divide the cube edges 1:3. Then you can repeat this periodically to get a triply periodic polyhedron where at each vertex exactly 12 triangles meet. This object is already invariant under the translations along the cube diagonals, so the quotient surface is an polyhedral surface of genus 4 (Euler!), tiled with 24 equilateral triangles, still 12 at each vertex. We can unwrap this package into the hyperbolic plane by replacing each Euclidean triangle with a hyperbolic 30º triangle:

The triangles corresponding to individual anti-prisms are colored with the same color, and the waist polygons of the anti-prisms become the shaded geodesics. That they close on the antiprism after passing through six triangles enforces the identification pattern of this 24-gon, indicated by letters. If you continue this, you can check that this hyperbolic version is in fact invariant not only under the 90º rotation about the center, but under the 30º rotation. In particular, we have also a 120º rotation, and the quotient of the surface by the generated cyclic group is a sphere tiled with 8 triangles — an octahedron, as is not hard to see. The same surface is also the basis for a famous triply periodic minimal surface. The inner 12-gon corresponds to the following minimal piece:

Copies of it can be assembled to larger portions, like so

or so:

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