User:Richard Pinch/sandbox-11

Honeycomb

"A symmetrical subdivision of a three-dimensional manifold into a number of polyhedral cells all alike, each rotation that is a symmetry of a cell being also a symmetry of the entire configuration."

A regular honeycomb is described by a Schläfli symbol $\{p,q,r\}$ denoting polyhedral cells that are Platonic solids $\{p,q\}$, such that every face $\{p\}$ belongs to just two cells, and every edge to $r$ cells.

References

• H.S.M. Coxeter "Twisted honeycombs", Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics. No.4. American Mathematical Society (1970) ISBN 0-8218-1653-5 Zbl 0217.46502
• H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002

Schläfli symbol

A symbol encoding classes of polygons, polyhedra, polytopes and tessellations.

The symbol $\{p\}$ denotes a regular $p$-gon; the symbol $\{p,q\}$ a polyhedron with faces which are regular $p$-gons, $q$ of which meet at each vertex. The Platonic solids correspond to:

• tetrahedron: $\{3,3\}$;
• cube: $\{4,3\}$;
• octahedron: $\{3,4\}$;
• dodecahedron: $\{5,3\}$;
• icosahedron: $\{3,5\}$.

There are three plane tessellations: $\{3,6\}$, $\{4,4\}$, $\{6,3\}$. The dual solid or tessellation to $\{p,q\}$ is $\{q,p\}$.

The symbol $\{p,q,r\}$ denotes a polytope in four dimensions or a honeycomb.

References

• H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002