Order-4 square hosohedral honeycomb

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.

Order-4 square hosohedral honeycomb

Centrally projected onto a sphere
TypeDegenerate regular honeycomb
Schläfli symbol{2,4,4}
Coxeter diagrams
Cells{2,4}
Faces{2}
Edge figure{4}
Vertex figure{4,4}
DualOrder-2 square tiling honeycomb
Coxeter group[2,4,4]
PropertiesRegular

Images

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


Centered on pole

Centered on equator

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

Truncated order-4 square hosohedral honeycomb
Order-2 square tiling honeycomb
Truncated order-4 square hosohedral honeycomb

Partial tessellation with alternately colored cubes
Typeuniform convex honeycomb
Schläfli symbol{4,4}×{}
Coxeter diagrams

Cells{3,4}
Faces{4}
Vertex figureSquare pyramid
Dual
Coxeter group[2,4,4]
PropertiesUniform

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

References
    • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
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