Great prismated cubic honeycomb

Great prismated cubic honeycomb
Rank	4
Type	uniform
Space	Euclidean
Notation
Bowers style acronym	Gippich
Coxeter diagram	x4x3x4x ()
Elements
Cells	3N octagonal prisms, N great rhombicuboctahedra
Faces	6N+12N squares, 8N hexagons, 6N octagons
Edges	24N+24N
Vertices	24N
Vertex figure	Phyllic disphenoid, edge lengths √2 (3), √3 (2), and √2+√2 (1)
Measures (edge length 1)
Vertex density	$30\sqrt2-42 \approx 0.42640687119$
Dual cell volume	$\frac{7+5\sqrt2}6 \approx 2.34517796864$
Related polytopes
Army	Gippich
Regiment	Gippitch
Dual	Phyllic disphenoidal honeycomb
Conjugate	Quasiquasiprismated cubic honeycomb
Topological properties
Orientable	Yes
Properties
Symmetry	R4×2
Convex	Yes

The great prismated cubic honeycomb, or gippich, also known as the omnitruncated cubic honeycomb, or otch, is a convex uniform honeycomb. 2 great rhombicuboctahedra and 2 octagonal prisms join at each vertex of this honeycomb. As the name suggests, it is the omnitruncate of the R4 family.

This honeycomb can be alternated into a snub bicubic honeycomb, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create an edge-snub cubic honeycomb, cantic bisnub cubic honeycomb or great birhombitetrahedral honeycomb, which are nonuniform.

Vertex coordinates

The vertices of a great prismated cubic honeycomb of edge length 1 are given by all permutations of:

• $\left(±\frac12+(2+2\sqrt2)i,\,±\frac{1+\sqrt2}{2}+(2+2\sqrt2)j,\,±\frac{1+2\sqrt2}{2}+(2+2\sqrt2)k\right),$

Where i, j, and k range over the integers.

Gallery

• 

  Wireframe

• 

  Half symmetry

• 

  Orthogonal projection into Euclidean plane with V3 symmetry

Related polytopes

o4o3o4o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cubic honeycomb	chon	{4,3,4}
Truncated cubic honeycomb	tich	t{4,3,4}
Rectified cubic honeycomb	rich	r{4,3,4}
Bitruncated cubic honeycomb	batch	2t{4,3,4}
Rectified cubic honeycomb	rich	r{4,3,4}
Truncated cubic honeycomb	tich	t{4,3,4}
Cubic honeycomb	chon	{4,3,4}
Small rhombated cubic honeycomb	srich	rr{4,3,4}
Great rhombated cubic honeycomb	grich	tr{4,3,4}
Small rhombated cubic honeycomb	srich	rr{4,3,4}
Great rhombated cubic honeycomb	grich	tr{4,3,4}
Small prismated cubic honeycomb = Cubic honeycomb	chon	t0,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t0,1,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t0,1,3{4,3,4}
Great prismated cubic honeycomb	gippich	t0,1,2,3{4,3,4}

External links

• Klitzing, Richard. "gippich".

• Wikipedia Contributors. "Omnitruncated cubic honeycomb".
• Binnendyk, Eric. "Category 5: Greater Truncates" (#99).