# Apeir operation

(Redirected from Apeir)

The **apeir operation** is an operation on finite realized regular polytopes. The apeir of a polytope , denoted , generates a tiling with zigzags as faces and as a vertex figure. The apeir operation preserves dimension, but it increases rank by 1.

## Definition[edit | edit source]

Let P be a polytope with generating mirrors . Then if w is point reflection about the initial vertex of P , apeir P is defined as . The new initial vertex is placed at the intersection ; i.e. the center of P .

## Examples[edit | edit source]

### 1D[edit | edit source]

### 2D[edit | edit source]

- Triangle --> Petrial hexagonal tiling
- Square --> Petrial square tiling
- Hexagon --> Petrial triangular tiling

### 3D[edit | edit source]

- Tetrahedron --> Apeir tetrahedron
- Cube --> Apeir cube
- Octahedron --> Apeir octahedron
- Petrial tetrahedron --> Petrial apeir tetrahedron
- Petrial cube --> Petrial apeir cube
- Petrial octahedron --> Petrial apeir octahedron

The apeir of other finite regular polygons or polyhedra of full rank will result in dense polytopes. The apeir operation distributes over blends, so the apeir of the blend of two polytopes is the blend of each of their apeirs.

## Properties[edit | edit source]

- The apeir of a polytope is discrete iff the polytope has a rational coordinates in any dimension.
- A polytope is pure iff its apeir is pure.
- The Petrial and apeir operations commute.
- A polytope of rank 3 or greater has a Petrial iff its apeir has a Petrial.