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 In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

of dimensions is a

skew polygon Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do ...

in which every consecutive sides (but no ) belongs to one of the

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

s. The Petrie polygon of a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

is the regular polygon itself; that of a

regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

is a skew polygon such that every two consecutive sides (but no three) belongs to one of the

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

s. 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 it wer ...

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 ar ...

of the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of the polygon, and the number of sides, , is the

Coxeter number 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 ...

of the

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

. 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

Egyptologist Egyptology (from ''Egypt'' and Greek , '' -logia''; ar, علم المصريات) is the study of ancient Egyptian history, language, literature, religion, architecture and art from the 5th millennium BC until the end of its native religiou ...

Hilda Hilda is one of several female given names derived from the name ''Hild'', formed from Old Norse , meaning 'battle'. Hild, a Nordic-German Bellona, was a Valkyrie who conveyed fallen warriors to Valhalla. Warfare was often called Hild's Game. ...

and

Flinders Petrie Sir William Matthew Flinders Petrie ( – ), commonly known as simply Flinders Petrie, was a British Egyptologist and a pioneer of systematic methodology in archaeology and the preservation of artefacts. He held the first chair of Egyp ...

. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability. In periods of intense concentration he could answer questions about complicated four-dimensional objects by ''visualizing'' them. He first noted the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra: :One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedral; infinite but free of false vertices. When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex, and one consisting of hexagons, four at each vertex. In 1938 Petrie collaborated with Coxeter,

Patrick du Val Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named aft ...

, and H.T. Flather to produce ''

The Fifty-Nine Icosahedra ''The Fifty-Nine Icosahedra'' is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forw ...

'' for publication.H. S. M. Coxeter,

, H.T. Flather, J.F. Petrie (1938) ''The Fifty-nine Icosahedra'',

University of Toronto The University of Toronto (UToronto or U of T) is a public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution ...

studies, mathematical series 6: 1–26 Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

''. The idea of Petrie polygons was later extended to

semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...

The Petrie polygons of the regular polyhedra

The regular duals, and , are contained within the same projected Petrie polygon. In the images of dual compounds on the right it can be seen that their Petrie polygons have rectangular intersections in the points where the edges touch the common midsphere. The Petrie polygons of the Kepler–Poinsot polyhedra are

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

s and decagrams . Infinite regular skew polygons (

apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to t ...

) can also be defined as being the Petrie polygons of the regular tilings, having angles of 90, 120, and 60 degrees of their square, hexagon and triangular faces respectively. : petrie_polygons_of_regular_tilings

Infinite regular skew polygons also exist as Petrie polygons of the regular hyperbolic tilings, like the

order-7 triangular tiling In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . Hurwitz surfaces The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is ...

, : :

Order-7 triangular tiling petrie polygon

The Petrie polygon of regular polychora (4-polytopes)

The Petrie polygon for the regular polychora can also be determined, such that every three consecutive sides (but no four) belong to one of the polychoron's cells.

The Petrie polygon projections of regular and uniform polytopes

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

Hypercubes

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

of dimension ''n'' has a Petrie polygon of size 2''n'', which is also the number of its

facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

.
So each of the (''n'' − 1)-cubes forming its

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

has ''n'' − 1 sides of the Petrie polygon among its edges.

Irreducible polytope families

This table represents Petrie polygon projections of 3 regular families (

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...

), and the

exceptional Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian sym ...

''E''_''n'' which generate semiregular and uniform polytopes for dimensions 4 to 8.

Notes

References

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, H. S. M. (1947, 63, 73) ''Regular Polytopes'', 3rd ed. New York: Dover, 1973. (sec 2.6 ''Petrie Polygons'' pp. 24–25, and Chapter 12, pp. 213–235, ''The generalized Petrie polygon '') * Coxeter, H.S.M. (1974) ''Regular complex polytopes''. Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons * Ball, W. W. R. and H. S. M. Coxeter (1987) ''Mathematical Recreations and Essays'', 13th ed. New York: Dover. (p. 135) * Coxeter, H. S. M. (1999) ''The Beauty of Geometry: Twelve Essays'', Dover Publications *

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

, Egon Schulte (2002) ''Abstract Regular Polytopes'',

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...

. * Steinberg, Robert,''ON THE NUMBER OF SIDES OF A PETRIE POLYGON'', 201

External links

* * * * * * * {{DEFAULTSORT:Petrie Polygon Polytopes