# Halving

Halving is an operation on regular polyhedra with square faces.

## Geometric description

Starting with a polytope π, to halve it we can first construct π . π  starts with the same vertices as π and we add edges to it connecting vertices that are opposite each other in π. That is two vertices incident on the same face, that are not connected by an edge in π, are connected by an edge in π . Then we add faces to π . For each vertex x  of π we add a face to π  made from the vertices that were adjacent to x  in π and the edges that connect them.

π  is then either a polyhedron or a compound of two identical polyhedra. The halving is an arbitrary connected component of π .

## Definition

### Distinguished generators

Let π be a polyhedron of SchlΓ€fli type {4,q}, with distinguished generators ${\displaystyle \langle \rho _{0},\,\rho _{1},\,\rho _{2}\rangle }$. Then the halving of π, denoted ${\displaystyle {\mathcal {P}}^{\eta }}$, is the polyhedron given by the distinguished generators ${\displaystyle \langle \rho _{0}\rho _{1}\rho _{0},\,\rho _{2},\,\rho _{1}\rangle }$.

## Properties

• Halving maintains regularity, meaning that the halving of a polytope is always regular.
• The faces of a halved polytope are equivalent to the vertex figure of the original polytope.
• For an abstract polytope {4,qh} then its halving is {q,q}{h}{}. Analogously, for a realized polytope, π, the Petrial of its halving is the blend of the dyad with the hole of π.

## Relationship to alternation

Halving is very closely related to alternation. Both can be seen as a generalization of the same simple concept. Any to any polytope with a bipartite skeleton can be alternated, while any regular polyhedron with square faces can be halved. If alternation is taken to replace digons with edges then the two produce the same results on the intersection of their domain.

Comparison of polytopes which can be alternated and halved
Can be halved Can be alternated
Cube Yes Yes
Triangular duocomb Yes No
Hexagonal prism No Yes
Tesseract No Yes
Tetrahedron No No

Unlike alternation, halving is deterministic on its domain, meaning there is exactly 1 halving for every regular polyhedron with square faces.

## Generalized halving

There also exists a generalized halving operation ${\displaystyle \eta _{k,m}}$ defined on polyhedra with ${\displaystyle 2(k+m)}$-gonal faces rather than on square faces.[1] The ${\displaystyle \eta _{k,m}}$-halving of a regular polyhedron ${\displaystyle (\rho _{0},\rho _{1},\rho _{2})}$ is defined using distinguished generators as ${\displaystyle ((\rho _{0}\rho _{1})^{k}\rho _{0},\rho _{2},(\rho _{1}\rho _{0})^{m-1}\rho _{1})}$. With this notation, the normal halving operation ${\displaystyle \eta =\eta _{1,1}}$, and unlike the normal halving ${\displaystyle \eta _{k,m}}$-halving is not always self-dual - instead it is related by ${\displaystyle \delta \eta _{k,m}\delta =\eta _{m,k}}$. Generalized halving is also closely related to the generalized skewing operation ${\displaystyle \sigma _{k,m}}$.

Generalized halving operations define further relationships between regular polyhedra. For example, applying ${\displaystyle \eta _{1,2}}$ to the muoctahedron gives the square tiling.

## References

1. β McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes. Cambridge University Press. ISBN 0-521-81496-0.