Order-4 square hosohedral honeycomb

From Wikipedia, the free encyclopedia
  (Redirected from Cubic prismatic slab honeycomb)
Jump to: navigation, search
Order-4 square hosohedral honeycomb
Order-4 square hosohedral honeycomb-sphere.png
Centrally projected onto a sphere
Type Degenerate regular honeycomb
Schläfli symbol {2,4,4}
Coxeter diagrams CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Cells {2,4} Spherical square hosohedron2.png
Faces {2}
Edge figure {4}
Vertex figure {4,4}
Square tiling uniform coloring 1.png
Dual {4,4,2}
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 {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.

Images[edit]

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

Order-4 square hosohedral honeycomb-stereographic.png
Centered on pole
Order-4 square hosohedral honeycomb-stereographic2.png
Centered on equator

Related honeycombs[edit]

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

Truncated order-4 square hosohedral honeycomb[edit]

Order-2 square tiling honeycomb
Truncated order-4 square hosohedral honeycomb
Cubic semicheck.png
Partial tessellation with alternately colored cubes
Type uniform convex honeycomb
Schläfli symbol {4,4}×{}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Cells {3,4} Hexahedron.png
Faces {4}
Vertex figure Square pyramid
Dual
Coxeter group [2,4,4]
Properties Uniform

The {2,4,4} honeycomb can be truncated as t{2,4,4} or {}×{4,4}, Coxeter diagram CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png, 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, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png, consists of infinite square pyramids and infinite tetrahedrons, between 2 square tilings.

See also[edit]

References[edit]