# Apeir operation

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

## Definition

Let P  be a polytope with generating mirrors ${\displaystyle \langle \rho _{0},\rho _{1},\dots ,\rho _{n}\rangle }$. Then if w  is point reflection about the initial vertex of P , apeir P  is defined as ${\displaystyle \langle w,\rho _{0},\rho _{1},\dots ,\rho _{n}\rangle }$. The new initial vertex is placed at the intersection ${\displaystyle \rho _{0}\cap \rho _{1}\cap \dots \cap \rho _{n}}$; i.e. the center of P .

## Examples

### 3D

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

• 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.