Webbed space

Webbed space
Type	Dual uniform honeycomb
Coxeter–Dynkin diagrams	Template:CDD
Cell	Square bipyramids
Faces	Triangles
Space group Fibrifold notation	PmTemplate:Overlinem (221) 4−:2
Coxeter group	${\tilde {C}}_{3}$, [4,3,4]
vertex figures	Template:CDD, Template:CDD
Dual	Rectified cubic honeycomb
Properties	Cell-transitive, Face-transitive

The square bipyramidal honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it an oblate octahedrille.

It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 elongated square pyramid cells. The the original cubic honeycomb walls are removed, joining pairs of square pyramids into elongated square bipyramids (octahedron). Its vertex and edge framework is identical to the hexakis cubic honeycomb.

There is one type of plane with faces: a flattended triangular tiling with half of the triangles as holes. These cut face-diagonally through the original cubes. There are also square tiling plane that exist as nonface holes passing through the centers of the octahedral cells.

Tiling plane	Square tiling "holes"	flattened triangular tiling
Symmetry	p4m, [4,4] (*442)	pmm, [∞,2,∞] (*2222)

Related honeycombs

It is dual to the rectified cubic honeycomb with octahedral and cuboctahedral cells:

See also

• Architectonic and catoptric tessellation

References

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• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, pp 292–298, includes all the nonprismatic forms)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49–56.