In geometry, a symmetrohedron is a high-symmetry polyhedron containing convex regular polygons on symmetry axes with gaps on the convex hull filled by irregular polygons.The name was coined by Craig S. Kaplan and George W. Hart.[1]
The trivial cases are the Platonic solids, Archimedean solids with all regular polygons. A first class is called bowtie which contain pairs of trapezoidal faces. A second class has kite faces. Another class are called LCM symmetrohedra.
Each symmetrohedron is described by a symbolic expression G(l; m; n; α). G represents the symmetry group (T,O,I). The values l, m and n are the multipliers ; a multiplier of m will cause a regular km-gon to be placed at every k-fold axis of G. In the notation, the axis degrees are assumed to be sorted in descending order, 5,3,2 for I, 4,3,2 for O, and 3,3,2 for T . We also allow two special values for the multipliers: *, indicating that no polygons should be placed on the given axes, and 0, indicating that the final solid must have a vertex (a zero-sided polygon) on the axes. We require that one or two of l, m, and n be positive integers. The final parameter, α, controls the relative sizes of the non-degenerate axis-gons.
Conway polyhedron notation is another way to describe these polyhedra, starting with a regular form, and applying prefix operators. The notation doesn't imply which faces should be made regular beyond the uniform solutions of the Archimedean solids.
These symmetrohedra are produced by a single generator point within a fundamental domains, reflective symmetry across domain boundaries. Edges exist perpendicular to each triangle boundary, and regular faces exist centered on each of the 3 triangle corners.
The symmetrohedra can be extended to euclidean tilings, using the symmetry of the regular square tiling, and dual pairs of triangular and hexagonal tilings. Tilings, Q is square symmetry p4m, H is hexagonal symmetry p6m.
Coxeter-Dynkin diagrams exist for these uniform polyhedron solutions, representing the position of the generator point within the fundamental domain. Each node represents one of 3 mirrors on the edge of the triangle. A mirror node is ringed if the generator point is active, off the mirror, and creates new edges between the point and its mirror image.