Halving

Halving is an operation on regular polyhedra with square faces.

Geometric description

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

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

Definition

Distinguished generators

Let 𝓟 be a polyhedron of Schläfli type {4,q}, with distinguished generators $\langle \rho _{0},\,\rho _{1},\,\rho _{2}\rangle$. Then the halving of 𝓟, denoted ${\mathcal {P}}^{\eta }$, is the polyhedron given by the distinguished generators $\langle \rho _{0}\rho _{1}\rho _{0},\,\rho _{2},\,\rho _{1}\rangle$.

Examples

• 

  The halved mucube is a regular polyhedron which can be constructed by halving the mucube. The image shows a section of the halved mucube overlain on the original mucube.

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.

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

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

Bibliography

• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.