Space-filling Rhombic Dodecahedra

The rhombic dodecahedron is a polyhedron with twelve rhombus-shaped faces, where each face has four sides of equal length. It is possible to construct the space-filling variant of the rhombic dodecahedron by arranging multiple such rhombic dodecahedra in a regular pattern so that they fill space without leaving any gaps. In his 1611 work on snowflakes titled “Strena seu de Nive Sexangula,” Johannes Kepler observed that honey bees utilize the geometric structure of rhombic dodecahedra to construct honeycombs. These honeycombs consist of a tessellation of cells, with each cell being a hexagonal prism topped with half a rhombic dodecahedron. I generated this structure in Grasshopper by utilizing the vertex coordinates of the polyhedron. Then, I defined three vectors to copy it in three directions. You can cut the output of the code and analyze the interior spaces, as seen in the animation.

This Grasshopper definition generates the Space-filling Rhombic Dodecahedra structure. The input parameters are the number of cells in three directions. The output of the definition is the solid 3d model. It has a structure with individual cells. Therefore, it is ready for 3d printing. The code is using native Grasshopper components. Thus, no add-ons are necessary for it to work.