Petrie polygon

The idea of Petrie polygons was later extended to semiregular polytopes.

Petrie polygon projections map the points into a regular 2n-gon or lower order regular polygons.

The Petrie polygon of regular polychora (4-polytopes)

The Petrie polygon projections are most useful for visualization of polytopes of dimension four and higher.

The regular pentadecagon is the Petrie polygon for one higher dimensional polytope, projected in a skew orthogonal projection:

The regular tetradecagon is the Petrie polygon for many higher dimensional polytopes, shown in these skew orthogonal projections, including:

The set of faces of any one of these two embeddings is the set of Petrie polygons of the other embedding.

These Petrie polygon (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.

A second projection takes the 2(n-1)-gon petrie polygon of the lower dimension, seen as a bipyramid, projected down the axis, with 2 vertices mapped into the center.

Examples include the Petrie polygons, polygonal paths of edges that divide a regular polytope into two halves, and seen as a regular polygon in orthogonal projection.

It is also the Petrie polygon for some higher dimensional polytopes with H symmetry, shown in orthogonal projections in the H Coxeter plane:

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within a 18-gonal regular polygon (called a Petrie polygon).

For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it.

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive sides (but no n) belong to one of the facets.

The two graphs shown are symmetric D and B Petrie polygon projections (2(n 1) and n dihedral symmetry) of the related polytope which can include overlapping edges and vertices.

This table represents Petrie polygon projections of 3 regular families (simplex, hypercube, orthoplex), and the Exceptional Lie group E which generate semiregular and uniform polytopes for dimensions 4 to 8.

Other types of nonface polygons associated with polyhedra and tessellations include Petrie polygons and facets (flat polygons formed by the polyhedron edges and vertices that are not faces of the polyhedron).

The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems - the vertices and edges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the Coxeter plane, yielding a Petrie polygon with h-fold rotational symmetry.