Weaire-Phelan structure

These are the base units of the Weaire-Phelan structure, as well as its periodic unit. The periodic units can be assembled into a 3d array without rotations.

In two dimensions, the subdivision of the plane into cells of equal area with minimum average perimeter is given by the hexagonal tiling. But how can space be partitioned into cells of equal volume with the least area of surface between them? This problem is referred to as the Kelvin problem.

One solution presented by Kelvin himself is called the Kelvin structure. His foam is based on the bitruncated cubic honeycomb, a convex uniform honeycomb formed by the truncated octahedron, a space-filling convex polyhedron with 6 square faces and 8 hexagonal faces.

It was not optimal, though: this honeycomb does not satisfy Plateau's laws.

In 1993 the Weaire–Phelan structure has been discovered. So far it has the smallest known surface area per cell and numerical experiments suggest that it is the optimal solution.

The periodic unit consists of 6 base units, where two units, one rotated by 90°, are joined into an axis base element. Those are then aligned on their respective cartesian axis and combined into the periodic unit.

When the periodic units are assembled into a 3d lattice, regular voids will show up that make up the other (conjugate) base structure of the Weaire-Phelan lattice.

The Weaire-Phelan structure can be used to create interesting patterns as in the Beijing National Aquatics Center. It is a regular packing of elements that is more efficient than the Kelvin structure. I plan to make it into an infill pattern for PrusaSlicer.

This model has been based on the construction of the Weaire-Phelan structure in Rhino by Tuğrul Yazar: https://www.designcoding.net/weaire-phelan-structure/