Hemitesseract

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

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.

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 ₂₄, a generalized cross polytope.

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. For example, the 2 in the first column of the second row indicates that there are 2 vertices in each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex.