Petrial

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The Petrial of the cube is the Petrial cube, shown above. Its faces are skew hexagons, labeled here in red, yellow, green, and blue.

A Petrial or the Petrie dual is an operation on polytopes. The Petrial is usually applied in the context of abstract or skew regular polytopes, since it is not closed on the planar polytopes. The Petrial of a polytope differs only in its facets, otherwise sharing all of their elements (the vertices edges etc.).

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, when applied to a polyhedron, its faces are the Petrie polygons of the original polytope.

Construction[edit | edit source]

The Petrial is classically an operation applied to polyhedra. To construct the Petrial of a base polytope, 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 do not repeat vertices, they will create the faces of the Petrial.

Any regular polyhedron has an associated Petrial, and these will be regular as well. The Petrial of an abstract regular polyhedron will almost always form another abstract regular polyhedron, but there are a few cases where the Petrie dual does not satisfy the intersection property; the regular map is an example. Whenever it is the case that Petrie dual of an abstract regular polyhedron is not a polyhedron, the Petrie polygons of the abstract regular polyhedron revisit vertices multiple times.

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.

Distinguished generators[edit | edit source]

An alternative definition of the Petrial can be given in terms of distinguished generators.

This definition is less broad than the previous, applying only to abstract regular polytopes and their symmetric realizations, however it is often easier to work with.

Properties[edit | edit source]

The Petrial exchanges the faces of a polyhedron with its Petrie polygons. If a polyhedron P  has Petrie polygons p , then its Petrial has faces p . Even stronger, for an abstract regular polyhedron defined by a Petrie polygon relation, {p , q }r , its Petrial is also defined by a Petrie polygon relation, {r , q }p .

This can be generalized even further; faces are the 0-holes of polyhedra, and Petrie polygons are the 1-zigzags. The Petrial exchanges n -holes with (n +1)-zigzags. For example the mucube which can be defined as the most general quotient of {4,6} with square holes {4,6∣4}. This has:

  • Square 0-holes (faces)
  • Square 1-holes
  • Infinite 1-zigzags (Petrie polygons)

Thus its Petrial will have:

  • Square 1-zigzags (Petrie polygons)
  • Square 2-zigzags
  • Infinite 0-holes (faces)

and it is exactly the quotient of {∞,4} with square 1 and 2 zigzags: {∞,6}4,4.

Higher ranks[edit | edit source]

For ranks greater than 3, the Petrial is defined as the polytope formed by taking the Petrial of the polytope's vertex figure.[1] 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.

This can be expressed in terms of the distinguished generators as:

For ranks higher than 3, the Petrial is not guaranteed to produce another regular polytope and only works in a select few cases.

Polychora[edit | edit source]

For polychora the formula for the Petrial can be expressed as:

The cells of a Petrial have a tendency to have holes. For example both the Petrial tesseract and the Petrial cubic honeycomb have cells with non-trivial square holes. It can be shown by manipulating the generators that the holes of P  are faces of P π  .

For example McMullen (2004) shows that if the first number of a polychoron's Schläfli type is odd then the Petrial does not exist. This theorem is sufficient to eliminate all the convex and star polychora other than the tesseract, which has a known Petrial.

Furthermore a polychoron has a Petrial iff its kappa has a Petrial.

Ranks > 4[edit | edit source]

For higher ranks the Petrial exists iff its vertex figure has a Petrial.

External links[edit | edit source]

References[edit | edit source]

  1. McMullen (2004)

Bibliography[edit | edit source]

  • McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete Computational Geometry.