geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a skew polygon is a closed

polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

. It is a

figure Figure may refer to: General *A shape, drawing, depiction, or geometric configuration *Figure (wood), wood appearance *Figure (music), distinguished from musical motif * Noise figure, in telecommunication * Dance figure, an elementary dance patt ...

similar to a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

except its vertices are not all

coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...

. While a polygon is ordinarily defined as a plane figure, the edges and vertices of a skew polygon form a

space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

. Skew polygons must have at least four vertices. The ''interior''

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 ...

and corresponding area measure of such a polygon is not uniquely defined. Skew infinite polygons (apeirogons) have vertices which are not all colinear. A zig-zag skew polygon or

antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...

atic polygonRegular complex polytopes, p. 6 has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygons in 3 dimensions (and regular skew apeirogons in two dimensions) are always zig-zag.

Skew polygons in three dimensions

A regular skew polygon is a faithful symmetric realization of a polygon in dimension greater than 2. In 3 dimensions a regular skew polygon has vertices alternating between two parallel planes. A regular skew -gon can be given a

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

as a ''blend'' of a

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

and an orthogonal

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

.Abstract Regular Polytopes, p.217 The symmetry operation between sequential vertices is

glide reflection In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation. Bec ...

. Examples are shown on the uniform square and pentagon antiprisms. The star antiprisms also generate regular skew polygons with different connection order of the top and bottom polygons. The filled top and bottom polygons are drawn for structural clarity, and are not part of the skew polygons.

Petrie polygon 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 ...

s are regular skew polygons defined within regular polyhedra and polytopes. For example, the five

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s have 4-, 6-, and 10-sided regular skew polygons, as seen in these

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 we ...

s with red edges around their respective projective envelopes. The tetrahedron and the octahedron include all the vertices in their respective zig-zag skew polygons, and can be seen as a digonal antiprism and a triangular antiprism respectively.

Regular skew polygon as vertex figure of regular skew polyhedron

A regular skew polyhedron has regular polygon faces, and a regular skew polygon

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

. Three infinite regular skew polyhedra are space-filling in 3-space; others exist in 4-space, some within the uniform 4-polytopes.

Regular skew polygons in four dimensions

In 4 dimensions, a regular skew polygon can have vertices on a

Clifford torus In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles and (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Cliffo ...

and related by a Clifford displacement. Unlike zig-zag skew polygons, skew polygons on double rotations can include an odd-number of sides. The

s of the regular 4-polytopes define regular zig-zag skew polygons. The Coxeter number for each

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 ref ...

symmetry expresses how many sides a Petrie polygon has. This is 5 sides for a

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...

, 8 sides for a

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

and 16-cell, 12 sides for a

24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...

, and 30 sides for a

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...

and

600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...

. When orthogonally projected onto the Coxeter plane, these regular skew polygons appear as regular polygon envelopes in the plane. The ''n''-''n'' duoprisms and dual duopyramids also have 2''n''-gonal Petrie polygons. (The

is a 4-4 duoprism, and the 16-cell is a 4-4 duopyramid.)

Citations

References

* p. 25 * "Skew Polygons (Saddle Polygons)" §2.2 * * Coxeter, H.S.M.; ''Regular complex polytopes'' (1974). Chapter 1. ''Regular polygons'', 1.5. Regular polygons in n dimensions, 1.7. ''Zigzag and antiprismatic polygons'', 1.8. ''Helical polygons''. 4.3. ''Flags and Orthoschemes'', 11.3. ''Petrie polygons'' * Coxeter, H. S. M. ''Petrie Polygons.''

Regular Polytopes ''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...

, 3rd ed. New York: Dover, 1973. (sec 2.6 ''Petrie Polygons'' pp. 24–25, and Chapter 12, pp. 213–235, ''The generalized Petrie polygon'') * (1st ed, 1957) 5.2 The Petrie polygon . *

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...

: ''On the total curvature of knots'', Ann. Math. 52 (1950) 248–257. * J.M. Sullivan: ''Curves of finite total curvature'', ArXiv:math.0606007v2

External links