Space-filling 3D honeycomb lattices with a laser cutter

The 3D equivalent of a 2D tiling or tesselation is called a honeycomb. (Actual bee honeycombs consist of rhombic dodecahedra, and correspond to the densest possible sphere packing.)

If you have access to a laser cutter (or other way of making precise cuts, or even a 3D printer in a pinch), you can buld your own honeycomb structures. These are attractive design elements and a fun construction toy. This post outlines the general approach and gives two specific examples.

3D honeycombs consist of packed arrangements of polyhedra (tetrahedra, cubes, etc.). To design laser-cut pieces:

• Build honeycombs by arranging flat pieces according to the faces of these polyhedra. (Generate a set of pieces for each face of the polyhedra in the honeycomb.)
• Join polyhedron faces in 3D by using additional (flat) notched pieces. Where the edges of different polyhedra meet, calculate the dihedral angles of the joining faces. Create notched joining pieces with these angles.

For most polyhedra, one can simple look up the dihedral angles. But, there is also some nice geometric reasoning behind this: Joining polyhedral faces with flat connecting pieces is equivalent to constructing the dual honeycomb. 

A dual honeycomb is conceptually similar to a dual tiling or dual polyhedron. In a dual honeycomb, one places a vertex at the center of every polyhedron, and connects these vertices with edges that pass perpendicularly through each face in the original honeycomb. For example, the dual of the rhombic dodecahedral honeycomb (used by bees) is the tetrahedral-octahedral honeycomb. 

 

Here are two examples: the rhombic dodecahedral honeycomb and the rectified cubic honeycomb.

 

Rhombic dodecahedral honeycomb

 

The rhombic dodecahedra consists of 12 rhombic facets with the long diagonal $\sqrt{2}$ times the length of the short diagonal, creating an acute angle of $\operatorname{arccos}(1/3)$ and obtuse angle of $\pi-\operatorname{arccos}(1/3)$.

The rhombic dodecahedra has two types of vertices: (a) one where the acute angles of 4 rhombi meet, and (b) one where the obtuse angles of 3 rhombi meet. In a honeycomb, (a) connects the 4-rhombus points from 6 dodecahedra, and (b) connects the 3-rhombus points from 4 dodecahedra.

This hints at the structure of the dual honeycomb: we should expect a dual lattice that consits of (1) polyhedra with six vertices, and (2) polyhedra with 4 vertices. Indeed, these correspond to the octahedra and tetrahedra of the tetrahedral-octahedral honeycomb.

The faces of each of the polyhedra in the dual lattice are equilateral triangles. The rhombic dodecahedral honeycomb can therefore be built from the rhombi described above, and equilateral triangles. (we also could have simply looked up that the rhombic dodecahedron has a dihedral angle of 120°)

 

 

Rectified cubic honeycomb

 

The rectified cubic honeycomb (cool rendering) can be constructed from cuboctahedra and octrahedra. The facets of these polyhedra are squares and equilateral triangles, and are easy to construct.

The dual lattice (joining pieces) is slightly trickier. Each angle joins two cuboctahedra and one octahedron, which have dihedral angles $\operatorname{arccos}(−1/\sqrt{3})$ and $\operatorname{arccos}(−​1/3)$, respectively (125.26°, and 109.5°). This corresponds to a triangle with angles 62.6°, 62.6°, and 54.7°. The dual honeycomb is the square bipyramidal honeycomb.

The rectified cubic honeycomb can therefore be build with three pieces: squares, equilateral triangles, and the above-described isosceles triangle.