Hemitesseract
| Hemitesseract (hemi-4-cube) | |
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| Type | Regular projective 4-polytope |
| Schläfli symbol | {4,3,3}/2 or {4,3,3}4 |
| Cells | 4 {4,3}  |
| Faces | 12 {4} |
| Edges | 16 |
| Vertices | 8 |
| Vertex figure | Tetrahedron |
| Petrie polygon | Square |
| Dual | hemi-16-cell |
In abstract geometry, a hemitesseract is an abstract, regular 4-polytope, existing in real projective space RP3.[1] It contains one k-face for each pair of opposite k-faces in the tesseract.
It has four cubic cells, 12 square faces, 16 edges, and 8 vertices. It has an unexpected property that every cell is in contact with every other cell on two faces, and every cell contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets.
The following images show the four pairs of opposite cubes in the tesseract (each with two highlighted pairs of opposite squares). Below are the corresponding cubes in the hemitesseract.
| tesseract | pairs of opposite tesseract cells | |||
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| hemitesseract | hemitesseract cells | |||
| representations in cubic hull | ||||
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These images show two ways to assign the vertices of a hemitesseract to those of a cube. | ||||
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As a graph
The skeleton is the complete bipartite graph K4,4. The images show four white and four gray vertices (for even and odd binary weight of the tesseract coordinates). Each vertex of one color is connected to all vertices of the other (but to none of the same).
It is also the regular complex polygon 2{4}4, a generalized cross polytope.
As a configuration
The configuration matrix below represents the hemitesseract. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole hemitesseract. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[2] For example, the 2 in the first column of the second row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex.
See also
References
- ↑ McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective Regular Polytopes", Abstract Regular Polytopes (1st ed.), Cambridge University Press, pp. 162–165, ISBN 0-521-81496-0
- ↑ Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover. p. 12, §1.8 Configurations.