Rectified tesseractic honeycomb explained

bgcolor=#e7dcc3 colspan=2	quarter cubic honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
Type	Uniform 4-honeycomb
Family	Quarter hypercubic honeycomb
Schläfli symbol	r r r q
Coxeter-Dynkin diagram	= = = =
4-face type	h, h₃,
Cell type	, t₁,
Face type
Edge figure	Square pyramid
Vertex figure	Elongated ×
Coxeter group	{\tilde{C}}₄ = [4,3,3,4] {\tilde{B}}₄ = [4,3<sup>1,1</sup>] {\tilde{D}}₄ = [3<sup>1,1,1,1</sup>]
Dual
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the rectified tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16-cell facets at the original vertices. Its vertex figure is an octahedral prism, ×.

It is also called a quarter tesseractic honeycomb since it has half the vertices of the 4-demicubic honeycomb, and a quarter of the vertices of a tesseractic honeycomb.^[1]

References

Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 https://books.google.com/books?id=fUm5Mwfx8rAC&dq=%22quarter+cubic+honeycomb%22+q%7B4%2C3%2C4%7D&pg=PA318
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
o4x3o3o4o, o3o3o *b3x4o, x3o3x *b3o4o, x3o3x *b3o *b3o - rittit - O87
Book: Conway JH, Sloane NJH . 1998 . Sphere Packings, Lattices and Groups . 3rd . 0-387-98585-9 . registration .