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 ...

, an octagon (from the

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, t, which alternates two types of edges. A truncated octagon, t is a hexadecagon, . A 3D analog of the octagon can be the

rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...

with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.

Properties of the general octagon

The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", ''Forum Geometricorum'' 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.

Skew octagon

A skew octagon 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 ...

with 8 vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A ''skew zig-zag octagon'' has vertices alternating between two parallel planes. A regular skew octagon is

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

with equal edge lengths. In 3-dimensions it will be a zig-zag skew octagon and can be seen in the vertices and side edges of a

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular, it is a sem ...

with the same D_4d, ⁺,8symmetry, order 16.

Petrie polygons

The regular skew octagon is the

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 ...

for these higher-dimensional regular and

uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...

s, shown in these skew

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 ...

s of in A₇, B₄, and D₅

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 ...

Symmetry of octagon

The ''regular octagon'' has Dih₈ symmetry, order 16. There are 3 dihedral subgroups: Dih₄, Dih₂, and Dih₁, and 4 cyclic subgroups: Z₈, Z₄, Z₂, and Z₁, the last implying no symmetry. On the regular octagon, there are 11 distinct symmetries. John Conway labels full symmetry as r16.John H. Conway, Heidi Burgiel,

Chaim Goodman-Strauss Chaim Goodman-Strauss (born June 22, 1967 in Austin TX) is an American mathematician who works in convex geometry, especially aperiodic tiling. He is on the faculty of the University of Arkansas and is a co-author with John H. Conway of ''The Sym ...

, (2008) The Symmetries of Things, (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r16 and no symmetry is labeled a1. The most common high symmetry octagons are p8, an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8, an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ...

of each other and have half the symmetry order of the regular octagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g8 subgroup has no degrees of freedom but can seen as

directed edge In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pai ...

Uses of octagons

The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa, Basilica of San Vitale (in Ravenna, Italia), Castel del Monte, Apulia, Castel del Monte (Apulia, Italia), Florence Baptistery, Zum Friedefürsten Church (Germany) and a number of octagonal churches in Norway. The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, Aachen, Palatine Chapel, has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral. Architects such as John Andrews (architect), John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, notably the Intelsat Headquarters in Washington D.C., Callam Offices in Canberra, and Octagon Offices in Parramatta, Australia.

Other uses

zont 8 ugolnik.jpg, Umbrellas often have an octagonal outline. Afghancarpet1.jpg, The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif. Eixample.svg, The street & block layout of Barcelona's Eixample district is based on non-regular octagons Janggipieces.jpg, Janggi uses octagonal pieces. Revolving lottery machine,kaitenshiki-cyusenki,japan.JPG, Japanese lottery machines often have octagonal shape. MUTCD R1-1.svg, Stop sign used in English language, English-speaking countries, as well as in most European countries Stop hand nuvola.svg, An icon of a stop sign with a hand at the middle. Bagua-name-earlier.svg, The trigrams of the Taoism, Taoist ''bagua'' are often arranged octagonally Octagonal footed gold cup from the Belitung shipwreck, ArtScience Museum, Singapore - 20110618-01.jpg, Famous octagonal gold cup from the Belitung shipwreck Shimer College class 1995 octagonal table.jpg, Classes at Shimer College are traditionally held around octagonal tables Labyrinthe de la cathédrale de Reims.svg, The Labyrinth of the Reims Cathedral with a quasi-octagonal shape. GameCube Analog Stick.jpg, The movement of the analog stick(s) of the Nintendo 64 controller, the GameCube controller, the Wii Nunchuk and the Classic Controller is restricted by a rotated octagonal area, allowing the stick to move in only eight different directions.

Derived figures

File:Tiling Semiregular 4-8-8 Truncated Square.svg, The truncated square tiling has 2 octagons around every vertex.
File:Octagonal prism.png, An octagonal prism contains two octagonal faces.
File:Octagonal antiprism.png, An octagonal antiprism contains two octagonal faces.
File:Great rhombicuboctahedron.png, The truncated cuboctahedron contains 6 octagonal faces.
File:Omnitruncated cubic honeycomb2.png, The omnitruncated cubic honeycomb

Related polytopes

The ''octagon'', as a truncated

, is first in a sequence of truncated hypercubes: As an expansion (geometry), expanded square, it is also first in a sequence of expanded hypercubes:

References

External links

Octagon Calculator
With interactive animation {{Polygons 8 (number) Constructible polygons Polygons by the number of sides Elementary shapes