Snub trihexagonal antiprismatic honeycomb

Snub trihexagonal antiprismatic honeycomb
Rank	4
Type	Isogonal
Space	Euclidean
Notation
Coxeter diagram	s∞o2s6s3s
Elements
Cells	12N irregular tetrahedra, 3N rhombic disphenoids, 2N triangular gyroprisms, N hexagonal gyroprisms
Faces	12N+12N+12N scalene triangles, 2N triangles, N hexagons
Edges	3N+6N+6N+6N+6N+6N
Vertices	12N
Vertex figure	Mirror-symmetric triangular-pentagonal orthobigyrowedge
Measures (based on great rhombitrihexagonal prismatic honeycomb of edge length 1)
Edge lengths	Diagonals of original squares (3N+6N+6N+6N): $\sqrt2 ≈ 1.41421$
	Edges of triangles (6N): $\sqrt3 ≈ 1.73205$
	Edges of hexagons (6N): $\frac{\sqrt2+\sqrt6}{2} ≈ 1.93185$
Related polytopes
Dual	Floret pentagonal antitegmatic honeycomb
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	V3❘W2+
Convex	Yes
Nature	Tame

The snub trihexagonal antiprismatic honeycomb is an isogonal honeycomb that consists of hexagonal gyroprisms, triangular gyroprisms, rhombic disphenoids, and irregular tetrahedra. 2 hexagonal gyroprisms, 2 triangular gyroprisms, 2 rhombic disphenoids, and 8 irregular tetrahedra join at each vertex. It can be obtained through the process of alternating the great rhombitrihexagonal prismatic honeycomb. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{13+\sqrt{57}}}{4}$ ≈ 1:1.13330.

External links

• Klitzing, Richard. "s∞o2s3s6s".