Regular skew apeirohedron


In geometry, a regular skew apeirohedron is an infinite skew polyhedron. They have either skew regular faces or skew regular vertex figures.

History

In 1926 John Flinders Petrie took the concept of a regular skew polygons, polygons whose vertices are not all in the same plane, and extended it to polyhedra. While apeirohedra are typically required to tile the 2-dimensional plane, Petrie considered cases where the faces were still convex but were not required to lie flat in the plane, they could have a skew polygon vertex figure.
Petrie discovered two regular skew apeirohedra, the mucube and the muoctahedron. Harold Scott MacDonald Coxeter derived a third, the mutetrahedron, and proved that these three were complete. Under Coxeter and Petrie's definition, requiring convex faces and allowing a skew vertex figure, the three were not only the only skew apeirohedra in 3-dimensional Euclidean space, but they were the only skew polyhedra in 3-space as there Coxeter showed there were no finite cases.
In 1967 Garner investigated regular skew apeirohedra in hyperbolic 3-space with Petrie and Coxeters definition, discovering 31 regular skew apeirohedra with compact or paracompact symmetry.
In 1977 Grünbaum generalized skew polyhedra to allow for skew faces as well. Grünbaum discovered an additional 23 skew apeirohedra in 3-dimensional Euclidean space and 3 in 2-dimensional space which are skew by virtue of their faces. 12 of Grünbaum's polyhedra were formed using the blending operation on 2-dimensional apeirohedra, and the other 11 were pure, could not be formed by a non-trivial blend. Grünbaum conjectured that this new list was complete for the parameters considered.
In 1985 Dress found an additional pure regular skew apeirohedron in 3-space, and proved that with this additional skew apeirohedron the list was complete.

Regular skew apeirohedra in Euclidean 3-space

Petrie-Coxeter polyhedra

The three Euclidean solutions in 3-space are,, and. John Conway named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron.
  1. Mucube: : 6 squares about each vertex
  2. Muoctahedron: : 4 hexagons about each vertex
  3. Mutetrahedron: : 6 hexagons about each vertex
Coxeter gives these regular skew apeirohedra with extended chiral symmetry which he says is isomorphic to his abstract group. The related honeycomb has the extended symmetry.

Grünbaum-Dress polyhedra

Skew honeycombs

There are 3 regular skew apeirohedra of full rank, also called regular [|skew honeycombs], that is skew apeirohedra in 2-dimensions. As with the finite skew polyhedra of full rank, all three of these can be obtained by applying the Petrie dual to planar polytopes, in this case the three regular tilings.
Alternatively they can be constructed using the apeir operation on regular polygons. While the Petrial is used the classical construction, it does not generalize well to higher ranks. In contrast, the apeir operation is used to construct higher rank skew honeycombs.
The apeir operation takes the generating mirrors of the polygon, and, and uses them as the mirrors for the vertex figure of a polyhedron, the new vertex mirror is then a point located where the initial vertex of the polygon. The new initial vertex is placed at the intersection of the mirrors and. Thus the apeir polyhedron is generated by.

Blended apeirohedra

For any two regular polytopes, and, a new polytope can be made by the following process:
  • Start with the Cartesian product of the vertices of with the vertices of.
  • Add edges between any two vertices and if there is an edge between and in and an edge between and in.
  • Similarly add faces to every set of vertices all incident on the same face in both and.
  • Repeat as such for all ranks of proper elements.
  • From the resulting polytope, select one connected component.
For regular polytopes the last step is guaranteed to produce a unique result. This new polytope is called the blend of and and is represented.
Equivalently the blend can be obtained by positioning and in orthogonal spaces and taking composing their generating mirrors pairwise.
Blended polyhedra in 3-dimensional space can be made by blending 2-dimensional polyhedra with 1-dimensional polytopes. The only 2-dimensional polyhedra are the 6 honeycombs :

Pure apeirohedra

A polytope is considered pure if it cannot be expressed as a non-trivial blend of two polytopes. A blend is considered trivial if it contains the result as one of the components. Alternatively a pure polytope is one whose symmetry group contains no non-trivial subrepresentation.
There are 12 regular pure apeirohedra in 3 dimensions. Three of these are the Petrie-Coxeter polyhedra:
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