In geometry, the order-4 square hosohedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 2,4,4}. It has 4 square hosohedra { 2,4} 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
The { 2,4,4} honeycomb can be truncated as t{ 2,4,4} or { }×{ 4,4}, 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)
Honeycombs (geometry)