Gyrochoron

A gyrochoron, also known as a step tegum, is an isochoric polychoron formed as the dual to the step prism. It can have any number of cells as long as it is more than or equal to 5, while still remaining isochoric. The cell of a gyrochoron always contains two-fold rotational symmetry, and all of its faces contain mirror symmetry. Higher types of face symmetry are possible if n and d are both even or if n and d are coprime and d² is equivalent to 1 mod n, allowing for faces with rectangular symmetry (such as the 15-4 gyrochoron). The only possible vertex figures for gyrochora are trapezohedra, tetragonal disphenoids, rhombic disphenoids, and phyllic disphenoids, which are related to the possible cell types of step prisms.

There exist gyrochora with any amount of cells starting from five, often more than one. This is in stark contrast to 3D space, where all isohedral shapes must have evenly many sides, and where bipyramids and trapezohedra are often the only examples of an isohedral polyhedron with a given face count.

External links

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".