# CRF (InstanceAttribute, 5)

### From Hi.gher. Space

An n-dimensional polytope is said to be **c**onvex, **r**egular-**f**aced (CRF) if and only if:

- It is convex;
- All its 2-dimensional elements are regular polygons.

The second requirement implies that all edge lengths are equal.

## Examples

All regular and uniform polytopes in all dimensions are CRF.

### 2D

In 2D, the CRF polytopes are exactly the regular polygons.

### 3D

In 3D, the CRF polytopes consist of the 5 Platonic solids, the 13 Archimedean solids, the 92 Johnson solids, and the infinite families of regular prisms and antiprisms. These are the only CRF polytopes in 3D.

### 4D

In 4D, the regular polychora and uniform polychora are CRF. Besides these, there are also a large number of other 4D polytopes that fall under the definition of CRF. As of 2016, the class of CRF polytopes has not yet been completely enumerated. The CRF polychora discovery project is an ongoing effort to discover and classify all 4D CRF polytopes.

As of 2016, the lower bound on the total number of 4D CRF polytopes (excluding members of infinite families, a few of which are known) was found to be at least in the millions, a large part of which arise from CRF diminishings of the 600-cell and, to a lesser extent, the augmentations of the duoprisms.

### 5D and beyond

A number of families of CRF polytopes are known to exist across all dimensions, and some families of CRF polytopes are also known to exist only up to a certain dimension. However, as of 2016, not much else is known about the scope of CRF polytopes in 5D and beyond, except that there are likely to be very large numbers of them.