Order-4 square hosohedral honeycomb explained

bgcolor=#e7dcc3 colspan=2	Order-4 square hosohedral honeycomb
Centrally projected onto a sphere
Type	Degenerate regular honeycomb
Schläfli symbol
Coxeter diagrams
Cells
Faces
Edge figure
Vertex figure
Dual	Order-2 square tiling honeycomb
Coxeter group	[2,4,4]
Properties	Regular

In geometry, the order-4 square hosohedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol . It has 4 square hosohedra around each edge. In other words, it is a packing of infinitely tall square columns. It is a degenerate honeycomb in Euclidean space, but can be seen as a projection onto the sphere. Its vertex figure, a square tiling is seen on each hemisphere.

Images

Stereographic projections of spherical projection, with all edges being projected into circles.

Related honeycombs

It is a part of a sequence of honeycombs with a square tiling vertex figure:

Truncated order-4 square hosohedral honeycomb

bgcolor=#e7dcc3 colspan=2	Order-2 square tiling honeycomb Truncated order-4 square hosohedral honeycomb Partial tessellation with alternately colored cubes
Type	uniform convex honeycomb
Schläfli symbol	×
Coxeter diagrams
Cells
Faces
Vertex figure	Square pyramid
Dual
Coxeter group	[2,4,4]
Properties	Uniform

The honeycomb can be truncated as t or ×, Coxeter diagram, seen as a layer of cubes, partially shown here with alternately colored cubic cells. Thorold Gosset identified this semiregular infinite honeycomb as a cubic semicheck.

The alternation of this honeycomb,, consists of infinite square pyramids and infinite tetrahedrons, between 2 square tilings.

See also

• Order-6 triangular hosohedral honeycomb
• Order-7 tetrahedral honeycomb
• List of regular polytopes

References

• The Beauty of Geometry: Twelve Essays (1999), Dover Publications,, (Chapter 10, Regular Honeycombs in Hyperbolic Space)