Snub rectified cubic honeycomb

Snub rectified cubic honeycomb
Rank	4
Type	Isogonal
Space	Euclidean
Notation
Bowers style acronym	Serch
Coxeter diagram	s4s3s4o ()
Elements
Cells	3N tetragonal disphenoids, 12N sphenoids, N pyritohedral icosahedra, N snub cubes
Faces	8N triangles, 12N+12N isosceles triangles, 24N scalene triangles, 3N squares
Edges	6N+12N+12N+24N
Vertices	12N
Vertex figure	Mirror-symmetric tridiminished icosahedron
Measures (based on alternating unit uniform great rhombated cubic honeycomb)
Edge lengths	Edges from diagonals of original squares (6N+12N): ${\sqrt {2}}\approx 1.41421$
	Edges of equilateral triangles (24N): ${\sqrt {3}}\approx 1.73205$
	Edges of squares (12N): ${\sqrt {2+{\sqrt {2}}}}\approx 1.84776$
Related polytopes
Army	Serch
Regiment	Serch
Dual	Semistellated sphenoidal honeycomb
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	R4×I+
Convex	Yes
Nature	Tame

The snub rectified cubic honeycomb or serch is an isogonal honeycomb that consists of snub cubes, pyritohedral icosahedra, tetragonal disphenoids, and sphenoids. 2 snub cubes, 1 pyritohedral icosahedron, 1 tetragonal disphenoid, and 4 sphenoids join at each vertex. It can be obtained through the process of alternating the great rhombated cubic honeycomb.

Although it cannot be made uniform, a version of this honeycomb with uniform snub cubes exists. It has two different edge lengths with a ratio of 1:${\sqrt {\frac {2}{t}}}\approx 1.04277$, where $t$ is the tribonacci constant, though this is not the lowest possible ratio between the longest and shortest edges.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:a ≈ 1:1.02551, where a is the second largest real root of 43x⁶-138x⁴+128x²-32=0, and is therefore a near-miss uniform polyhedral honeycomb. In this variant, none of the cells are regular, though the sphenoids and tetragonal disphenoids become identical to each other.

External links

• Klitzing, Richard. "serch".
• Wikipedia contributors. "Alternated cantitruncated cubic honeycomb".