Honeycombs

Uniform honeycombs were useful in finding new polyhedra. By definition the vertices of a uniform honeycomb are identical, so by removing a subset of polyhedra from the honeycomb can result in an isogonal polyhedron.

This method was probably how previous researchers found many of their polyhedra, so few new polyhedra have been found in this way. The exceptions to this are the alternated cubic honeycombs (octet, gyetoh and gytoh). Computer software was used to search these honeycombs for new polyhedra and many were found, but mainly by splitting octahedra into two square pyramids. In comparison, only two new polyhedra were discovered among the other cubic honeycombs and then only by splitting cuboctahedra into two triangular cupolae.

What was useful was the discovery of a number of new honeycombs found by including the cylindrical polyhedra. Cylinder₄ (Cy₄) is a square cylinder with triangle strip sides and cylinder₈ (Cy₈) is an octangular cylinder with alternating triangle and square strip sides. Neither can be broken down into simpler polyhedra. Other cylinders (hexagonal, octagonal and dodecagonal) also lead to new honeycombs, but none of these result in new polyhedra.

The uniform honeycombs have been broken down into four sections. The first three section are for the cubic, the alternated cubic and the prismatic honeycombs. The final section is for other honeycombs that include non-uniform polyhedra such as the cylinders and the pyramid₄ (Py₄: the square pyramid or half octahedron), the cupola₃ (Cp₃: the triangular cupola or half cuboctahedron) and the bicupola₄ (Bc₄: the square orthobicupola).

The table shown below shows the breakdown of each honeycomb into its components. On each line the numbers indicate how many of a particular polyhedron meet at each vertex. Where there is a tiling listed, it indicates that there is a plane through the honeycomb of this form.