The Petrie polygon of the dodecahedron is a skew decagon. Seen from the solid's 5-fold symmetry axis it looks like a regular decagon. Every pair of consecutive sides belongs to one pentagon (but no triple does).
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides (but no n) 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.[1] Petrie polygons are named for mathematician John Flinders Petrie.
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. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, h, 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.[2]
^Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] (Definition: paper 13, Discrete groups generated by reflections, 1933, p. 161)
^Gorini, Catherine A. (2000), Geometry at Work, MAA Notes, vol. 53, Cambridge University Press, p. 181, ISBN 9780883851647
In geometry, a Petriepolygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides (but no n) belongs to one...
structural clarity, and are not part of the skew polygons. Petriepolygons are regular skew polygons defined within regular polyhedra and polytopes. For...
realize the importance of the warped polygon that now bears his name, he was also skilled as a draftsperson. Petrie was born on April 26, 1907, in Hampstead...
of a polygon pp. 275-278) Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141 Coxeter, Regular polytopes, 12.4 Petriepolygon, pp....
example shown below. This decomposition can be seen as 6 of 24 faces in a Petriepolygon projection plane of the tesseract. The list (sequence A006245 in the...
the Petriepolygon for three 8-dimensional polytopes with E8 symmetry, shown in orthogonal projections in the E8 Coxeter plane. It is also the Petrie polygon...
triangular antiprism) have regular skew hexagons as petriepolygons. The regular skew hexagon is the Petriepolygon for these higher dimensional regular, uniform...
the tridecagrams are compound figures. The regular tridecagon is the Petriepolygon 12-simplex: Gleason, Andrew Mattei (March 1988). "Angle trisection,...
that are proportioned in golden ratio. The regular decagon is also the Petriepolygon of the regular dodecahedron and icosahedron, and it is the largest face...
122. It is also the Petriepolygon for the grand 120-cell and great stellated 120-cell. A dodecagram is a 12-sided star polygon, represented by symbol...
regular stars and not compound figures. The regular heptadecagon is the Petriepolygon for one higher-dimensional regular convex polytope, projected in a skew...
equal. More generally regular skew polygons can be defined in n-space. Examples include the Petriepolygons, polygonal paths of edges that divide a regular...
4 squares and 3 sets of 8 rhombs. This decomposition is based on a Petriepolygon projection of an 8-cube, with 28 of 1792 faces. The list OEIS: A006245...
solid polygons, a polygon may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygons and...
divided into 36: 4 sets of 9 rhombs. This decomposition is based on a Petriepolygon projection of a 9-cube, with 36 of 4608 faces. The list OEIS: A006245...
since 7 is prime all solutions, q=1..6, are polygons. Regular skew tetradecagons exist as Petriepolygon for many higher-dimensional polytopes, shown...
(vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. The regular pentadecagon is the Petriepolygon for some higher-dimensional...
is the Petriepolygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including: Constructible Polygon John H...
vertices on the surface of a cylinder. Regular infinite skew polygons exist in the Petriepolygons of the affine and hyperbolic Coxeter groups. They are constructed...
10 600-cells Great polygons 2 squares x 3 4 rectangles x 4 4 hexagons x 4 12 decagons x 6 100 irregular hexagons x 4 Petriepolygons 1 pentagon x 2 1 octagon...
have Möbius–Kantor polygons as faces, each with eight vertices and eight complex equilateral triangles as edges, whose Petriepolygons form regular octagons...
Matthew Flinders Petrie FRS FBA ((1853-06-03)3 June 1853 – (1942-07-28)28 July 1942), commonly known as simply Sir Flinders Petrie, was a British Egyptologist...
these) are orthogonally projected onto the Coxeter plane, yielding a Petriepolygon with h-fold rotational symmetry. For root systems, no root maps to zero...
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