In
geometryGeometry 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....
, the
gyroelongated dipyramids are an infinite set of
polyhedraA polyhedron is often defined as a geometric solid with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is...
, constructed by elongating an
n-agonal
bipyramidAn n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
by inserting an
n-agonal
antiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
between its congruent halves.
Two members of the set can be
deltahedraA deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
, that is, constructed entirely of equilateral triangles: the
gyroelongated square dipyramidIn geometry, the gyroelongated square dipyramid is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves. It is a deltahedron....
, a
Johnson solidIn geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
, and the
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids....
, a
Platonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
. The gyroelongated triangular dipyramid can be made with equilateral triangles, but is not a deltahedron because it has coplanar faces, i.e.
Discussion
Ask a question about 'Gyroelongated dipyramid' Start a new discussion about 'Gyroelongated dipyramid' Answer questions from other users |
| Set of gyroelongated dipyramids |
|
| Faces |
4n triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s |
| Edges |
6n |
| Vertices |
2n+2 |
Symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
|
Dnd |
| Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
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truncated trapezohedraAn n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices....
|
| Properties |
convex |
In
geometryGeometry 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....
, the
gyroelongated dipyramids are an infinite set of
polyhedraA polyhedron is often defined as a geometric solid with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is...
, constructed by elongating an
n-agonal
bipyramidAn n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
by inserting an
n-agonal
antiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
between its congruent halves.
Two members of the set can be
deltahedraA deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
, that is, constructed entirely of equilateral triangles: the
gyroelongated square dipyramidIn geometry, the gyroelongated square dipyramid is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves. It is a deltahedron....
, a
Johnson solidIn geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
, and the
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids....
, a
Platonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
. The gyroelongated triangular dipyramid can be made with equilateral triangles, but is not a deltahedron because it has coplanar faces, i.e. is not strictly convex. The other members can be constructed with isosceles triangles.
Forms
- Gyroelongated triangular dipyramid - dual triangular truncated trapezohedron
- Gyroelongated square dipyramid
In geometry, the gyroelongated square dipyramid is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves. It is a deltahedron....
- dual square truncated trapezohedronThe square truncated trapezohedron is the second in an infinite series of truncated trapezohedron polyhedra. It has 8 pentagon and 2 square faces.This polyhedron can be constructed by taking a tetragonal trapezohedron and truncating the polar axis vertices....
- Gyroelongated pentagonal dipyramid icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids....
- dual pentagonal truncated trapezohedron dodecahedronIn geometry, a dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices and 30 edges...
- Gyroelongated hexagonal dipyramid - dual hexagonal truncated trapezohedron
The hexagonal truncated trapezohedron is the fourth in an infinite series of truncated trapezohedron polyhedra. It has 12 pentagon and 2 hexagon faces....
- ...
- n-agonal gyroelongated dipyramid - duals truncated trapezohedron
An n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices....
External links