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An n-agonal bipyramid or dipyramid (also known as a Deltahedron) is 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....
 formed by joining an n-agonal pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
 and its mirror image
Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
 base-to-base.

The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves.

The face-transitive bipyramids are the dual polyhedra
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 of the uniform prisms
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
 and will generally have isosceles triangle faces.

A bipyramid can be projected on a sphere or globe
Globe

A globe is a three-dimensional scale Model of Earth or other spheroid celestial body such as a planet, star, or moon. It may also refer to a spherical representation of the celestial sphere, showing the apparent positions of the stars in the sky ...
 as n equally spaced lines of longitude
Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
 going from pole
Geographical pole

A geographical pole , is either of two points on the surface of a spinning planet or other spinning body, at 90 degrees from its equator, at one of the two points where the Axis of rotation around which the body spins meets the surface of the body....
 to pole, and bisected
Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line , which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors....
 by a line around the equator
Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
.

Bipyramid 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....
s, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry
List of spherical symmetry groups

List of symmetry groups on the sphere Spherical symmetry groups are also called point groups in three dimensions. This article is about Point_groups_in_three_dimensions#Finite_isometry_groups....
 Dnh.

Equilateral triangle bipyramids
Only 3 kinds of bipyramids can have all edges of the same length (which implies that all faces are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s, and thus the bipyramid is a deltahedron
Deltahedron

A deltahedron is a polyhedron whose face s are all equilateral triangles. The name is taken from the Greek language majuscule delta , which has the shape of an equilateral triangle....
): the triangular
Triangular dipyramid

In geometry, the triangular dipyramid is the first in the infinite set of face-transitive dipyramids. It is the Dual polyhedron of the triangular prism with 6 isosceles triangle faces....
, Square, and pentagonal
Pentagonal dipyramid

In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive dipyramids.The set of dipyramids is the dual polyhedron of the Prism s....
 bipyramids.






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An n-agonal bipyramid or dipyramid (also known as a Deltahedron) is 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....
 formed by joining an n-agonal pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
 and its mirror image
Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
 base-to-base.

The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves.

The face-transitive bipyramids are the dual polyhedra
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 of the uniform prisms
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
 and will generally have isosceles triangle faces.

A bipyramid can be projected on a sphere or globe
Globe

A globe is a three-dimensional scale Model of Earth or other spheroid celestial body such as a planet, star, or moon. It may also refer to a spherical representation of the celestial sphere, showing the apparent positions of the stars in the sky ...
 as n equally spaced lines of longitude
Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
 going from pole
Geographical pole

A geographical pole , is either of two points on the surface of a spinning planet or other spinning body, at 90 degrees from its equator, at one of the two points where the Axis of rotation around which the body spins meets the surface of the body....
 to pole, and bisected
Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line , which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors....
 by a line around the equator
Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
.

Bipyramid 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....
s, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry
List of spherical symmetry groups

List of symmetry groups on the sphere Spherical symmetry groups are also called point groups in three dimensions. This article is about Point_groups_in_three_dimensions#Finite_isometry_groups....
 Dnh.

Equilateral triangle bipyramids


Only 3 kinds of bipyramids can have all edges of the same length (which implies that all faces are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s, and thus the bipyramid is a deltahedron
Deltahedron

A deltahedron is a polyhedron whose face s are all equilateral triangles. The name is taken from the Greek language majuscule delta , which has the shape of an equilateral triangle....
): the triangular
Triangular dipyramid

In geometry, the triangular dipyramid is the first in the infinite set of face-transitive dipyramids. It is the Dual polyhedron of the triangular prism with 6 isosceles triangle faces....
, Square, and pentagonal
Pentagonal dipyramid

In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive dipyramids.The set of dipyramids is the dual polyhedron of the Prism s....
 bipyramids. The Square bipyramid with identical edges, or regular octahedron, counts among the Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s, while the triangular and pentagonal bipyramids with identical edges count among the Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s (J12 and J13).

Triangular Dipyramid

Triangular bipyramid
Tetragonal bipyramid
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....
Pentagonal Dipyramid

Pentagonal bipyramid


Forms

  1. Triangular bipyramid - 6 faces - dual triangular prism
    Triangular prism

    In geometry, a triangular prism or three-sided prism is a type of Prism ; it is a polyhedron made of a triangle base, a Translation copy, and 3 faces joining corresponding sides....
  2. Square bipyramid (the regular octahedron is a special case) - 8 faces - dual cube
    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. Pentagonal bipyramid - 10 faces - dual pentagonal prism
    Pentagonal prism

    In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....
  4. Hexagonal bipyramid
    Hexagonal bipyramid

    A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramid joined at their bases. The resulting solid has 12 triangular face , 8 vertex and 18 edges....
     - 12 faces - dual hexagonal prism
    Hexagonal prism

    In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce...
  5. Heptagonal bipyramid - 14 faces - dual heptagonal prism
  6. Octagonal bipyramid
    Octagonal bipyramid

    The octagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an octagonal bipyramid is to be face-transitive, all faces must be isosceles triangles....
     - 16 faces - dual octagonal prism
    Octagonal prism

    In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....
  7. Enneagonal bipyramid - 18 faces - dual enneagonal prism
  8. Decagonal bipyramid
    Decagonal bipyramid

    In geometry, a decagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an decagonal bipyramid is to be face-transitive, all faces must be isosceles triangles....
     - 20 faces - dual decagonal prism
    Decagonal prism

    In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....
  • ...n-agonal bipyramid - 2n faces - dual n-agonal prism
    Prism (geometry)

    In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....



3
4
5
6
8
10


Symmetry groups

If the base is regular and the line through the apexes
Apex (geometry)

In geometry, an apex is a descriptive label for a visual singular highest or most distant point or Vertex in an isosceles triangle, Pyramid or Cone , usually contrasting with the opposite side called the base....
 intersects the base at its center, the symmetry group
List of spherical symmetry groups

List of symmetry groups on the sphere Spherical symmetry groups are also called point groups in three dimensions. This article is about Point_groups_in_three_dimensions#Finite_isometry_groups....
 of the n-agonal bipyramid has dihedral symmetry Dnh of order 4n, except in the case of a regular octahedron, which has the larger 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....
 group Oh of order 48, which has 3 versions of D4h as subgroups. The rotation group
Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
 is Dn of order 2n, except in the case of a regular octahedron, which has the larger symmetry group O of order 24, which has 3 versions of D4 as subgroups.

Star bipyramids


Self-intersecting bipyramids exist with a star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
 central figure, defined by triangular faces connecting each polygon edge to these 2 points.

For example, a pentagrammic dipyramid is an isohedral star polyhedron
Star polyhedron

In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon giving it a star-like visual quality.There are two general kinds of star polyhedron:...
 composed of 10 intersecting isoceles triangles. It is the dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 to the pentagrammic prism
Pentagrammic prism

In geometry, the pentagrammic prism is one in an infinite set of nonconvex Prism formed by square sides and two regular star polygon caps, in this case two pentagrams....
.

External links

  • The Encyclopedia of Polyhedra
    • VRML
      VRML

      VRML is a standard file format for representing 3-D computer graphics interactive vector graphics, designed particularly with the World Wide Web in mind....
       models
      • Try: "dPn", where n=3,4,5,6... example "dP4" is an octahedron.