Compound of five octahemioctahedra
{{Short description|Polyhedral compound}}
{{Notability|Numbers|date=July 2019}}
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!bgcolor=#e7dcc3 colspan=2|Compound of five octahemioctahedra | |
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| bgcolor=#e7dcc3|Type | Uniform compound |
| bgcolor=#e7dcc3|Index | UC61 |
| bgcolor=#e7dcc3|Polyhedra | 5 octahemioctahedra |
| bgcolor=#e7dcc3|Faces | 40 triangles, 20 hexagons |
| bgcolor=#e7dcc3|Edges | 120 |
| bgcolor=#e7dcc3|Vertices | 60 |
| bgcolor=#e7dcc3|Symmetry group | icosahedral (Ih) |
| bgcolor=#e7dcc3|Subgroup restricting to one constituent | pyritohedral (Th) |
In geometry, this uniform polyhedron compound is a composition of 5 octahemioctahedra, in the same vertex arrangement as in the compound of 5 cuboctahedra. It could also be called an icosidisicosahedron.
Filling
There is some controversy on how to colour the faces of this polyhedron compound. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in). In addition, overlapping regions of coplanar faces can cancel each other out. Usage of the "neo filling" makes the compound of five octahemioctahedra a hollow polyhedron compound.{{cite web | url=http://polytope.net/hedrondude/polyhedra.htm | title=Uniform Polyhedra }}
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References
- {{citation|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=79|pages=447–457|year=1976|doi=10.1017/S0305004100052440|mr=0397554|issue=3|bibcode=1976MPCPS..79..447S|s2cid=123279687 }}.
{{reflist}}
- {{cite web |last=McCooey |first=Robert |title=Uniform Polyhedron Compounds |url=http://www.polytope.net/hedrondude/fivers.htm |website=Hedron Dude |access-date=24 June 2025}}