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| Hemicube | |
|---|---|
 | |
| Type | Abstract regular polyhedron Globallyprojective polyhedron |
| Faces | 3squares |
| Edges | 6 |
| Vertices | 4 |
| Euler char. | χ = 1 |
| Vertex configuration | 4.4.4 |
| Schläfli symbol | {4,3}/2 or{4,3}3 |
| Symmetry group | S4, order 24 |
| Dual polyhedron | hemi-octahedron |
| Properties | Non-orientable |
In abstractgeometry, ahemicube is anabstract,regular polyhedron,[citation needed] produced by cutting a cube in half with a plane that passes through 2 opposite corners and the midpoints of 2 edges.[1]
A hemicube is also sometimes called asquare hemiprism.[2]
It can be realized as aprojective polyhedron (atessellation of thereal projective plane by three quadrilaterals), which can be visualized by constructing the projective plane as ahemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
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.
From the point of view ofgraph theory theskeleton is atetrahedral graph, an embedding ofK4 (thecomplete graph with four vertices) on aprojective plane.
The hemicube should not be confused with thedemicube – 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, thehemicube is aquotient of the cube, while the vertices of thedemicube are asubset of the vertices of the cube.
The hemicube is thePetrie dual to the regulartetrahedron, with the four vertices, six edges of the tetrahedron, and threePetrie polygon quadrilateral faces. The faces can be seen as red, green, and blue edge colorings in thetetrahedral graph:
