A petrial polytope is a polytope that is considered to have skew elements, whose vertices do not lie on the same hyperplane. Petrials can exist in three dimensions or greater. Petrial polytopes share all of the vertices and edges of the base polytope.
The name can be considered an abbreviation of "Petrie dual," the operation that forms petrial polytopes. It can also be thought of as describing the fact that its facets are the Petrie polygons of the original polytope.
In three dimensions, a petrial polyhedron can be created from a base polyhedron. One takes all circuits of edges so that any two consecutive edges belong to a common face but no three consecutive edges do. If these circuits don't self-intersect, they will create the faces of the petrial.
Any regular polyhedron has an associated petrial, and these will be regular as well.
Petrials based on uniform polyhedra can be scaliform. For example, the petrial based on a uniform truncated tetrahedron can be said to have three skew dodecagons, which are clearly not isogonal (as facets) within the tetrahedral symmetry group.
The petrial of a regular polyhedron with distinguished generators is the regular polyhedron generated by the distinguished generators:
In dimensions greater than 3, the petrial is defined as the polytope formed by taking the Petrie dual of the polytope's vertex figure. For example, the tesseract has square faces and a tetrahedral vertex figure, so the petrial tesseract has square faces and a petrial tetrahedron vertex figure. In higher dimensions, the petrial is not guaranteed to produce another regular polytope and only works in a select few cases.
- Wikipedia Contributors. "Petrie dual"
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- ↑ McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete Computational Geometry: 20.