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Reciprocal constructions from 8-cell and 16-cell

The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular 16-cell, and the 16 half-integer vertices (± 1 / 2, ± 1 / 2, ± 1 / 2, ± 1 / 2) are the vertices of its dual, the tesseract (8-cell). The tesseract gives Gosset's construction of the 24-cell, equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction, equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described above). The analogous construction in 3-space gives the cuboctahedron (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract. This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.