Octagon G2 Symmetry

In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon.

A ''regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a hexadecagon, . A 3D analog of the octagon can be the rhombicuboctahedron 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 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 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 with the same D_4d, ⁺,8symmetry, order 16.

Petrie polygons

The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A₇, B₄, and D₅ Coxeter planes.

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, (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 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 edges.

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, 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, 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 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-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 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 square, is first in a sequence of truncated hypercubes:

As an expanded square, it is also first in a sequence of expanded hypercubes:

See also

*Bumper pool
* Hansen's small octagon
*Octagon house
* Octagonal number
* Octagram
*Octahedron, 3D shape with eight faces.
* Oktogon, a major intersection in Budapest, Hungary
* Rub el Hizb (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