# Scaliform polytope

A **scaliform polytope** is an isogonal polytope with a single edge length. Unlike uniforms, there is no requirement that elements are also isogonal. However, all elements must still be orbiform at least, i.e. having a single edge length and being inscribed in a hypersphere. E.g. the rhombic tiling, which would bow to the first 2 axioms, still does *not* obeye the third. The term "scaliform" originated in the online enthusiast community, with the term coined by Jonathan Bowers from the words "scale" and "uniform".

Scaliform polytopes are precisely the uniform polytopes up to and including 3D. However, in 4D and above there are scaliform polytopes that are not uniform. Most effort has concentrated on finding non-uniform scaliform polytopes that are also convex, but some nonconvex scaliforms have been studied too.

## Properties[edit | edit source]

Restricting discussion to finite planar polytopes or, more generally, to planar elements, it follows that faces are regular polygons always.

Using the spherical / parabolic / hyperbolic models of according geometries, embedded within euclidean space, the additional isogonality of elements already could be deduced from the other axioms by their then well-defined planarity. However it would not follow intrinsically only.

## 4D scaliforms[edit | edit source]

### Convex[edit | edit source]

Research into scaliforms was spawned by Richard Klitzing's 2000 discovery of the truncated tetrahedral alterprism (formerly called the truncated tetrahedral cupoliprism), the simplest convex, non-uniform scaliform polychoron.

There are only four known non-uniform convex scaliform polychora:

- truncated tetrahedral alterprism
- bi-icositetradiminished hexacosichoron
- prismatorhombisnub icositetrachoron
- swirlprismatodiminished rectified hexacosichoron

The cells of a convex scaliform polytope must be either convex uniform polyhedra or one of the Johnson solids with a circumscribed sphere.

### Non-convex[edit | edit source]

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. A notable case is the small pyramidic swirlprism which is non-convex but does not contain any self-intersections.

## 5D[edit | edit source]

In 5D, the duoantifastegiaprisms, also known as duoantiwedges, 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.

## 6D+[edit | edit source]

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.