Hemitesseract

Not to be confused with Demitesseract.

In abstract geometry, a hemitesseract is an abstract, regular 4-polytope, containing half the cells of a tesseract, existing in real projective space, RP³.^[1]

Hemitesseract (hemi-4-cube)
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

Realization

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.

In a cubic projection, the 4 cubic cells can be seen by selecting 3 of 4 sets of parallel edges. One is direct,and three are seen as cross-cubes. One of 6 square faces is shown yellow in each cube.	A projection inside a regular octagon, with two colors of vertices showing it as complete bipartite graph K4,4, and its 4 sets of 4 parallel edges.

As a graph

From the point of view of graph theory, the skeleton is a cubic graph with 8 diagonal central edges added.

It is also the complete bipartite graph K_4,4, and the regular complex polygon ₂{4}₄, a generalized cross polytope.^{[clarification needed]}

As a configuration

This configuration matrix 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.

${\displaystyle {\begin{bmatrix}{\begin{matrix}8&4&6&4\\2&16&3&3\\4&4&12&2\\8&12&6&4\end{matrix}}\end{bmatrix}}}$

See also

• Hemicube (geometry)