In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a Petrie polygon for a
regular polytope of dimensions is a
skew polygon in which every consecutive
sides (but no ) belongs to one of the
facets. The Petrie polygon of a
regular polygon is the regular polygon itself; that of a
regular polyhedron is a skew polygon such that every two consecutive sides (but no three) belongs to one of the
faces. Petrie polygons are named for mathematician
John Flinders Petrie.
For every regular polytope there exists an
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if i ...
onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the
Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
of the
symmetry group of the polygon, and the number of sides, , is the
Coxeter number of the
Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes.
Petrie polygons can be defined more generally for any
embedded graph. They form the faces of another embedding of the same graph, usually on a different surface, called the
Petrie dual.
History
John Flinders Petrie (1907–1972) was the son of
Egyptologists