Scaliform polytope

A scaliform polytope is an isogonal polytope that can be represented with only one edge length, without the consideration that the elements themselves need to be uniform. The elements of a scaliform polytope are all orbiform, however, so eg. only Johnson solids that have a circumscribed sphere are the valid convex cells beyond the uniform ones, and only regular polygons are valid faces. Infinite sets of scaliform polytopes can be created from the Cartesian product of a scaliform polytope and either a regular polygon or a 3D antiprism; these then exist in six dimensions or greater. In two dimensions, every scaliform polygon is regular, while in three dimensions, every scaliform polyhedron is uniform.

The concept was first considered in 2000 after Richard Klitzing's discovery of the truncated tetrahedral alterprism (formerly called the truncated tetrahedral cupoliprism), the simplest convex, non-uniform scaliform polychoron. Jonathan Bowers coined the name "scaliform" from the words "scale" and "uniform".

In 4D, there are only four convex scaliform polychora: the truncated tetrahedral alterprism, the bi-icositetradiminished hexacosichoron, the prismatorhombisnub icositetrachoron, and the swirlprismatodiminished rectified hexacosichoron, up to current knowledge. If we include non-convex cases, the total number of known scaliform polychora rises to an infinite amount, the hemiantiprisms form an infinite family, and at least 855 for non-prismatic cases, though like with the uniform polychora this list is not known to be complete.

In 5D, the duoantiwedges, also known as duoantifastegiaprisms, form an infinite family of scaliform polytera. There are also infinite families such as the polygonal disphenoids and duoprismatic cupoliprisms, though in each case only one member is convex. The number of total scaliform polytopes in 5D and higher is not yet known, as little research has been done.