Symmetrohedron

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.

Duals

I(*;2;3;e)	Pyritohedral

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.

Domain	Edges	Tetrahedral (3 3 2)		Octahedral (4 3 2)		Icosahedral (5 3 2)		Triangular (6 3 2)			Square (4 4 2)
		Symbol	Image	Symbol	Image	Symbol	Image	Symbol	Image	Dual	Symbol	Image	Dual
	1	T(1;*;*;e) T,		C, O(1;*;*;e)		I(1;*;*;e) D,		H(1;*;*;e) H,			Q(1;*;*;e) Q,
	1	T(*;1;*;e) dT,		O(*;1;*;e) O,		I(*;1;*;e) I,		H(*;1;*;e) dH,			Q(*;1;*;e) dQ,
	2	T(1;1;*;e) aT,		O(1;1;*;e) aC,		I(1;1;*;e) aD,		H(1;1;*;e) aH,			Q(1;1;*;e) aQ,
	3	T(2;1;*;e) tT,		O(2;1;*;e) tC,		I(2;1;*;e) tD,		H(2;1;*;e) tH,			Q(2;1;*;e) tQ,
	3	T(1;2;*;e) dtT,		O(1;2;*;e) tO,		I(1;2;*;e) tI,		H(1;2;*;e) dtH,			Q(1;2;*;e) dtQ,
	4	T(1;1;*;1) eT,		O(1;1;*;1) eC,		I(1;1;*;1) eD,		H(1;1;*;1) eH,			Q(1;1;*;1) eQ,
	6	T(2;2;*;e) bT,		O(2;2;*;e) bC,		I(2;2;*;e) bD,		H(2;2;*;e) bH,			Q(2;2;*;e) bQ,

• Near-miss Johnson solid
• Conway polyhedron notation

• Symmetrohedra on RobertLovesPi.net.
• Antiprism Free software that includes Symmetro for generating and viewing these polyhedra with Kaplan-Hart notation.