Digon

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Digon

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
a digon is a degenerate

Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
polygon

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
with two sides (edges) and two vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
.

A digon must be regular

Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
because its two edges are the same length. It has Schläfli symbol

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
.

a class="link1" onMouseover='showByLink("m6980341",this)' onMouseout='hide("m6980341")'href="http://www.absoluteastronomy.com/topics/Euclidean_geometry">Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
a digon is always degenerate. However, in spherical geometry

Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
a nondegenerate digon (with a nonzero interior area) can exist if the vertices are antipodal

Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diameter opposite it ? so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....
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Encyclopedia

In geometry

Geometry

Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
polygon

Polygon

Vertex (geometry)

Regular polygon

Schläfli symbol

In spherical tilings

In Euclidean geometry

Euclidean geometry

Spherical geometry

Antipodal point

Internal angle

In geometry, an interior angle is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon....
of the spherical digon vertex can be any angle between 0 and 180 degrees. Such a spherical polygon can also be called a lune

Lune (mathematics)

A lune is either of two figures, both shaped roughly like a crescent Moon. The word "lune" derives from luna, the Latin language word for Moon....
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One antipodal digon on the sphere.	Six antipodal digon faces on a hexagonal hosohedron Hosohedron An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol .... tiling on the sphere.

In polyhedra

A digon is considered degenerate face

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
of a polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
because it has no geometric area and overlapping edges, but it can sometimes have a useful topological existence in transforming polyhedra.

Any polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
can be topologically modified by replacing an edge with a digon. Such an operation adds one edge and one face to the polyhedron, although the result is geometrically identical. This transformation has no effect on the Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
(χ=V-E+F).

A digon face can also be created by geometrically collapsing a quadrilateral

Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
face by moving pairs of vertices to coincide in space. This digon can then be replaced by a single edge. It loses one face, two vertices, and three edges, again leaving the Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
unchanged.

Classes of polyhedra can be derived as degenerate forms of a primary polyhedron, with faces sometimes being degenerated into coinciding vertices. For example, this class of 7 uniform polyhedron

Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
with octahedral symmetry

Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
exist as degenerate forms of the great rhombicuboctahedron (4.6.8). This principle is used in the Wythoff construction

Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
.

4.4.4

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....

3.8.8

Truncated cube

The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....

3.4.3.4

Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....

4.6.6

Truncated octahedron

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

3.3.3.3

Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....

3.4.4.4

4.6.8

Digon

In spherical tilings

In polyhedra

See also

External links