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It has three square faces, six edges, and four vertices. It has an unexpected property that every face is in contact with every other face on two edges, and every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets.
269:– the hemicube is a projective polyhedron, while the demicube is an ordinary polyhedron (in Euclidean space). While they both have half the vertices of a cube, the
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where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
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by three quadrilaterals), which can be visualized by constructing the projective plane as a
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quadrilateral faces. The faces can be seen as red, green, and blue edge colorings in the
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367:(1st ed.), Cambridge University Press, pp.
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16:Abstract regular polyhedron with 3 square faces
265:The hemicube should not be confused with the
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277:of the cube, while the vertices of the
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285:of the vertices of the cube.
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47:Abstract regular polyhedron
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365:Abstract Regular Polytopes
235:From the point of view of
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258:with four vertices) on a
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212:It can be realized as a
19:Not to be confused with
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222:real projective plane
214:projective polyhedron
52:projective polyhedron
407:Projective polyhedra
293:The hemicube is the
107:Vertex configuration
247:, an embedding of
194:regular polyhedron
333:hemi-dodecahedron
307:tetrahedral graph
289:Related polytopes
245:tetrahedral graph
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338:hemi-icosahedron
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260:projective plane
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361:McMullen, Peter
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328:hemi-octahedron
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297:to the regular
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163:hemi-octahedron
158:Dual polyhedron
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119:Schläfli symbol
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303:Petrie polygon
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256:complete graph
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173:Non-orientable
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180:In abstract
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299:tetrahedron
295:Petrie dual
281:cube are a
208:Realization
92:Euler char.
354:References
273:cube is a
226:hemisphere
169:Properties
152:, order 24
349:Footnotes
50:Globally
401:Category
322:See also
275:quotient
267:demicube
241:skeleton
190:abstract
186:hemicube
182:geometry
82:Vertices
29:Hemicube
21:Demicube
369:162–165
220:of the
124:{4,3}/2
65:squares
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283:subset
188:is an
254:(the
243:is a
200:of a
198:faces
128:{4,3}
112:4.4.4
72:Edges
59:Faces
373:ISBN
279:demi
271:hemi
239:the
202:cube
184:, a
43:Type
216:(a
126:or
100:= 1
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249:K
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