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 ...
, 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 not ...
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
In geometry, 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 reg ...
.
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 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
In topological graph theory, the Petrie dual of an embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces.
The Petrie dual is also called the Petrial, and ...
.
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 ...
s
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. Th ...
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, ]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 ...
, 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 university, public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park (Toronto), Queen's Park. It was founded by royal charter in 1827 ...
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 ...
s.
The Petrie polygons of the regular polyhedra
The
regular duals, and , are contained within the same projected Petrie polygon.
In the images of
dual compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
The outer vertices of a compound can be connected ...
s on the right it can be seen that their Petrie polygons have rectangular intersections in the points where the edges touch the common
midsphere
In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex po ...
.
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 th ...
) 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](https://upload.wikimedia.org/wikipedia/commons/f/f4/Petrie_polygons_of_regular_tilings.png)
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 ...
, :
:
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
A
hypercube 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,
hypercube,
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 symm ...
''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 Pre ...
.
* Steinberg, Robert,''ON THE NUMBER OF SIDES OF A PETRIE POLYGON'', 201
See also
External links
*
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{{DEFAULTSORT:Petrie Polygon
Polytopes