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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 c ...
, 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 toge ...
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 D4d, +,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 ...
s.


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 pa ...
s.


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 The Tower of the Winds or the Horologion of Andronikos Kyrrhestes is an octagonal Pentelic marble clocktower in the Roman Agora in Athens that functioned as a ''horologion'' or "timepiece". It is considered the world's first meteorological stat ...
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, Italia),
Florence Baptistery The Florence Baptistery, also known as the Baptistery of Saint John ( it, Battistero di San Giovanni), is a religious building in Florence, Italy, and has the status of a minor basilica. The octagonal baptistery stands in both the Piazza del D ...
,
Zum Friedefürsten Church The Zum Friedefürsten Church (german: Rundkirche zum Friedefürsten) is a baroque Lutheran round church in Klingenthal, Saxony, south-eastern Germany. It has an octagonal floorplan and is the largest of its kind in Saxony. The church is the mos ...
(Germany) and a number of
octagonal churches in Norway An octagonal church has an octagonal (eight-sided polygon) architectural plan. The exterior and the interior (the nave) may be shaped as eight-sided polygon with approximately equal sides or only the nave is eight-sided supplemented by choir and ...
. The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal
apse In architecture, an apse (plural apses; from Latin 'arch, vault' from Ancient Greek 'arch'; sometimes written apsis, plural apsides) is a semicircular recess covered with a hemispherical vault or semi-dome, also known as an ''exedra''. In ...
of
Nidaros Cathedral Nidaros Cathedral ( no, Nidarosdomen / Nidaros Domkirke) is a Church of Norway cathedral located in the city of Trondheim in Trøndelag county. It is built over the burial site of Olav II of Norway, King Olav II (c. 995–1030, reigned 1015–102 ...
. Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, notably the
Intelsat Headquarters 3400 International Drive (also known as Intelsat Headquarters) is an office complex in the North Cleveland Park neighborhood of Washington, D.C. by the Van Ness metro station. Known for its futuristic and high-tech architecture, it was design ...
in Washington D.C., Callam Offices in Canberra, and Octagon Offices in
Parramatta Parramatta () is a suburb and major Central business district, commercial centre in Greater Western Sydney, located in the state of New South Wales, Australia. It is located approximately west of the Sydney central business district on the ban ...
, Australia.


Other uses

zont 8 ugolnik.jpg,
Umbrella An umbrella or parasol is a folding canopy supported by wooden or metal ribs that is usually mounted on a wooden, metal, or plastic pole. It is designed to protect a person against rain or sunlight. The term ''umbrella'' is traditionally use ...
s often have an octagonal outline. Afghancarpet1.jpg, The famous
Bukhara rug A Turkmen rug ( tk, Türkmen haly; or Turkmen carpet or Turkoman carpet) is a type of handmade floor-covering textile traditionally originating in Central Asia. It is useful to distinguish between the original Turkmen tribal rugs and the rugs pr ...
design incorporates an octagonal "elephant's foot" motif. Eixample.svg, The street & block layout of
Barcelona Barcelona ( , , ) is a city on the coast of northeastern Spain. It is the capital and largest city of the autonomous community of Catalonia, as well as the second most populous municipality of Spain. With a population of 1.6 million within ci ...
's
Eixample The Eixample (; ) is a district of Barcelona between the old city ( Ciutat Vella) and what were once surrounding small towns ( Sants, Gràcia, Sant Andreu, etc.), constructed in the 19th and early 20th centuries. Its population was 262,000 ...
district is based on non-regular octagons Janggipieces.jpg, Janggi uses octagonal pieces. Revolving lottery machine,kaitenshiki-cyusenki,japan.JPG, Japanese
lottery machine A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of ...
s often have octagonal shape. MUTCD R1-1.svg, Stop sign used in
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
-speaking countries, as well as in most
European countries The list below includes all entities falling even partially under any of the regions of Europe, various common definitions of Europe, geographical or political. Fifty generally recognised sovereign states, Kosovo with limited, but substantial, ...
Stop hand nuvola.svg, An icon of a stop sign with a hand at the middle. Bagua-name-earlier.svg, The trigrams of the
Taoist Taoism (, ) or Daoism () refers to either a school of philosophical thought (道家; ''daojia'') or to a religion (道教; ''daojiao''), both of which share ideas and concepts of Chinese origin and emphasize living in harmony with the ''Tao'' ...
''
bagua The bagua or pakua (八卦) are a set of eight symbols that originated in China, used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Each consists of three lines, each li ...
'' 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 The Belitung shipwreck (also called the Tang shipwreck or Batu Hitam shipwreck) is the wreck of an Arabian dhow which sank around 830 AD. The ship completed the outward journey from Arabia to China, but sank on the return journey from China, app ...
Shimer College class 1995 octagonal table.jpg, Classes at
Shimer College Shimer Great Books School (pronounced ) is a Great Books college that is part of North Central College in Naperville, Illinois. Prior to 2017, Shimer was an independent, accredited college on the south side of Chicago, with a history of being ...
are traditionally held around octagonal tables Labyrinthe de la cathédrale de Reims.svg, The
Labyrinth of the Reims Cathedral The Labyrinth of the Reims Cathedral was a church labyrinth installed on the floor of the nave of the Reims Cathedral. Structure The labyrinth was the shape of a complex square with cut corners and sides of . The paths were wide, separated ...
with a quasi-octagonal shape. GameCube Analog Stick.jpg, The movement of the
analog stick An analog stick (or analogue stick in British English), sometimes called a control stick or thumbstick, is an input device for a controller (often a game controller) that is used for two-dimensional input. An analog stick is a variation of a joy ...
(s) of the
Nintendo 64 controller The Nintendo 64 controller (model number: NUS-005) is the standard game controller for the Nintendo 64 home video game console. Manufactured and released by Nintendo on June 23, 1996, in Japan, in late 1996 in North America, and 1997 in Europe, ...
, the
GameCube controller The GameCube controller is the standard game controller for the GameCube home video game console, manufactured by Nintendo and launched in 2001. As the successor to the Nintendo 64 controller, it is the progression of Nintendo's controller des ...
, the
Wii Nunchuk The Wii Remote, also known Colloquialism, colloquially as the Wiimote, is the primary game controller for Nintendo's Wii home video game console. An essential capability of the Wii Remote is its Motion controller, motion sensing capability, which ...
and the
Classic Controller The is a game controller produced by Nintendo for the Wii home video game console. While it later featured some compatibility with the Wii U console, the controller was ultimately succeeded by the Wii U Pro Controller. In April 2014, Nintendo ...
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 In geometry, the truncated square tiling is a semiregular tiling, semiregular tiling by regular polygons of the Euclidean plane with one square (geometry), square and two octagons on each vertex (geometry), vertex. This is the only edge-to-edge ti ...
has 2 octagons around every vertex.
File:Octagonal prism.png, An
octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by rectangular sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron. Symmetry Images The octagonal prism can also ...
contains two octagonal faces.
File:Octagonal antiprism.png, An
octagonal antiprism In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each other ...
contains two octagonal faces.
File:Great rhombicuboctahedron.png, The
truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fac ...
contains 6 octagonal faces.
File:Omnitruncated cubic honeycomb2.png, The omnitruncated cubic honeycomb


Related polytopes

The ''octagon'', as a 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 ...
, is first in a sequence of truncated
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, ...
s: As an expanded square, it is also first in a sequence of expanded hypercubes:


See also

*
Bumper pool Bumper pool is a cue sport played on an rectangular (or sometimes octagonal) table fitted with two pockets and an array of fixed cushioned obstacles, called bumpers, within the interior of the table surface. Table Typically, bumper pool tabl ...
*
Hansen's small octagon The largest small octagon is the octagon that has the largest area among all convex octagons with unit diameter. The diameter of a polygon is the length of the longest segment joining two of its vertices. The exact value of the area of the larg ...
*
Octagon house Octagon houses were a unique house style briefly popular in the 1850s in the United States and Canada. They are characterised by an octagonal (eight-sided) plan, and often feature a flat roof and a veranda all round. Their unusual shape and app ...
*
Octagonal number An octagonal number is a figurate number that represents an octagon. The octagonal number for ''n'' is given by the formula 3''n''2 - 2''n'', with ''n'' > 0. The first few octagonal numbers are : 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 34 ...
*
Octagram In geometry, an octagram is an eight-angled star polygon. The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line". Deta ...
*
Octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, 3D shape with eight faces. * Oktogon, a major intersection in
Budapest Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population ...
,
Hungary Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Croatia a ...
*
Rub el Hizb The Rub-el-Hizb ( ar, ربع الحزب, '), also known as the Islamic Star, is an Islamic symbol. It is in the shape of an octagram, represented as two overlapping squares. It has been found on a number of emblems and flags. The main purpose of ...
(also known as Al Quds Star and as Octa Star) * Smoothed octagon


References


External links


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