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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. 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.
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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.