Quotient prism

A quotient prism is a polytope formed from the vertices of a polytope compound, where each individual element lies within the vertices of an orthogonal simplex. For example, the 12-4 step prism is based on the compound of three squares, where the vertices of each square lie within the vertices of an orthogonal equilateral triangle, hence it is a vertex-faceting of a triangular-dodecagonal duoprism. If the compound is isogonal, then the resulting quotient prism is isogonal. To designate a quotient prism, the terms n-gyroprism (as in tetragyroprism) or gyrosimplexism (as in gyrotetrahedronism) can be used.

Most quotient prisms do not have inversion symmetry, so they can be compounded with their inverses. The convex hull of the resulting figure is called a bi-n-gyroprism, such as the square dihedral bitrigyroprism, which is the convex hull of a tridiminished rectified hexateron (square dihedral trigyoprism) and its inverse.

The number of dimensions of a quotient prism is equal to (n+d-1), where n is the number of elements present in the compound and d is the dimension where the compound belongs to. Thus, the aforementioned 12-4 step prism has (3+2-1) = 4 dimensions.

The coordinates of a quotient prism lie on two independent sets; one describing the individual elements, and one describing the vertices of a simplex. Thus, the tridiminished rectified hexateron, which is based on the compound of three square dihedra, has the following coordinates, with three representing the squares, and the other two representing the vertices of an equilateral triangle:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,{\frac {\sqrt {6}}{6}}\right),$
• $\left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {6}}{12}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,-{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {6}}{12}}\right).$