Order-4 square hosohedral honeycomb
| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Order-4 square hosohedral honeycomb | |
| align=center
|colspan=2|300px | |
| bgcolor=#e7dcc3|Type | Degenerate regular honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {2,4,4} |
| bgcolor=#e7dcc3|Coxeter diagrams | {{CDD|node_1|2|node|4|node|4|node}} {{CDD|node_1|2|node|infin|node|2|node|infin|node}} |
| bgcolor=#e7dcc3|Cells | {2,4} 40px |
| bgcolor=#e7dcc3|Faces | {2} |
| bgcolor=#e7dcc3|Edge figure | {4} |
| bgcolor=#e7dcc3|Vertex figure | {4,4} 50px |
| bgcolor=#e7dcc3|Dual | Order-2 square tiling honeycomb |
| bgcolor=#e7dcc3|Coxeter group | [2,4,4] |
| bgcolor=#e7dcc3|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. 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.
| class=wikitable |
| align=center
|400px |
| align=center
|400px |
Related honeycombs
It is a part of a sequence of honeycombs with a square tiling vertex figure:
{{Square tiling vertex figure tessellations}}
= Truncated order-4 square hosohedral honeycomb =
| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Order-2 square tiling honeycomb | |
| bgcolor=#e7dcc3|Type | uniform convex honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {4,4}×{} |
| bgcolor=#e7dcc3|Coxeter diagrams | {{CDD|node_1|4|node|4|node|2|node_1}} {{CDD|node|4|node_1|4|node|2|node_1}} {{CDD|node_1|4|node|4|node_1|2|node_1}} |
| bgcolor=#e7dcc3|Cells | {3,4} 40px |
| bgcolor=#e7dcc3|Faces | {4} |
| bgcolor=#e7dcc3|Vertex figure | Square pyramid |
| bgcolor=#e7dcc3|Dual | |
| bgcolor=#e7dcc3|Coxeter group | [2,4,4] |
| bgcolor=#e7dcc3|Properties | Uniform |
The {2,4,4} honeycomb can be truncated as t{2,4,4} or {}×{4,4}, Coxeter diagram {{CDD|node_1|2|node_1|4|node|4|node}}, 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, {{CDD|node_h|2x|node_h|4|node|4|node}}, consists of infinite square pyramids and infinite tetrahedrons, between 2 square tilings.
See also
References
{{reflist}}
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, {{LCCN|99035678}}, {{ISBN|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space])