# Snub trihexagonal antiprismatic honeycomb

Snub trihexagonal antiprismatic honeycomb
Rank4
TypeIsogonal
SpaceEuclidean
Notation
Coxeter diagrams∞o2s6s3s
Elements
Cells12N irregular tetrahedra, 3N rhombic disphenoids, 2N triangular gyroprisms, N hexagonal gyroprisms
Faces12N+12N+12N scalene triangles, 2N triangles, N hexagons
Edges3N+6N+6N+6N+6N+6N
Vertices12N
Vertex figureMirror-symmetric triangular-pentagonal orthobigyrowedge
Measures (based on great rhombitrihexagonal prismatic honeycomb of edge length 1)
Edge lengthsDiagonals of original squares (3N+6N+6N+6N): ${\displaystyle \sqrt2 ≈ 1.41421}$
Edges of triangles (6N): ${\displaystyle \sqrt3 ≈ 1.73205}$
Edges of hexagons (6N): ${\displaystyle \frac{\sqrt2+\sqrt6}{2} ≈ 1.93185}$
Related polytopes
DualFloret pentagonal antitegmatic honeycomb
Abstract & topological properties
OrientableYes
Properties
SymmetryV3❘W2+
ConvexYes
NatureTame

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:${\displaystyle \frac{\sqrt{13+\sqrt{57}}}{4}}$ ≈ 1:1.13330.