Gyroelongated triangular bicupola

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Gyroelongated triangular bicupola
Gyroelongated triangular bicupola.png
Type Johnson
J43 - J44 - J45
Faces 2+3×6 triangles
6 squares
Edges 42
Vertices 18
Vertex configuration 6(3.4.3.4)
2.6(34.4)
Symmetry group D3
Dual polyhedron -
Properties convex, chiral
Net
Johnson solid 44 net.png

In geometry, the gyroelongated triangular bicupola is one of the Johnson solids (J44). As the name suggests, it can be constructed by gyroelongating a triangular bicupola (either J27 or the cuboctahedron) by inserting a hexagonal antiprism between its congruent halves.

Contents

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids , Archimedean solids , prisms , or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966. [1]

The gyroelongated triangular bicupola is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each square face on the bottom half of the figure is connected by a path of two triangular faces to a square face above it and to the right. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom square would be connected to a square face above it and to the left. The two chiral forms of J44 are not considered different Johnson solids.

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a: [2]

Related Research Articles

Triangular cupola

In geometry, the triangular cupola is one of the Johnson solids (J3). It can be seen as half a cuboctahedron.

Square cupola

In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (J4). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon.

Pentagonal rotunda

In geometry, the pentagonal rotunda is one of the Johnson solids (J6). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.

Elongated square cupola

In geometry, the elongated square cupola is one of the Johnson solids (J19). As the name suggests, it can be constructed by elongating a square cupola (J4) by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" removed.

Square gyrobicupola

In geometry, the square gyrobicupola is one of the Johnson solids (J29). Like the square orthobicupola (J28), it can be obtained by joining two square cupolae (J4) along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.

Gyroelongated pentagonal birotunda

In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (J48). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda by inserting a decagonal antiprism between its two halves.

Gyroelongated square bicupola

In geometry, the gyroelongated square bicupola is one of the Johnson solids (J45). As the name suggests, it can be constructed by gyroelongating a square bicupola by inserting an octagonal antiprism between its congruent halves.

Pentagonal cupola

In geometry, the pentagonal cupola is one of the Johnson solids (J5). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

Snub square antiprism

In geometry, the snub square antiprism is one of the Johnson solids (J85). A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.

Sphenocorona

In geometry, the sphenocorona is one of the Johnson solids (J86). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.

Triangular hebesphenorotunda

In geometry, the triangular hebesphenorotunda is one of the Johnson solids (J92).

Elongated triangular pyramid

In geometry, the elongated triangular pyramid is one of the Johnson solids (J7). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically self-dual.

Elongated triangular bipyramid

In geometry, the elongated triangular bipyramid or triakis triangular prism is one of the Johnson solids (J14), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid (J12) by inserting a triangular prism between its congruent halves.

Gyrobifastigium

In geometry, the gyrobifastigium is the 26th Johnson solid (J26). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

Gyroelongated pentagonal bicupola

In geometry, the gyroelongated pentagonal bicupola is one of the Johnson solids (J46). As the name suggests, it can be constructed by gyroelongating a pentagonal bicupola by inserting a decagonal antiprism between its congruent halves.

Elongated triangular cupola

In geometry, the elongated triangular cupola is one of the Johnson solids (J18). As the name suggests, it can be constructed by elongating a triangular cupola (J3) by attaching a hexagonal prism to its base.

Gyroelongated triangular cupola

In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J22). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J3). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.

Triangular orthobicupola

In geometry, the triangular orthobicupola is one of the Johnson solids (J27). As the name suggests, it can be constructed by attaching two triangular cupolas (J3) along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron. It is also a canonical polyhedron.

Elongated triangular gyrobicupola

In geometry, the elongated triangular gyrobicupola is one of the Johnson solids (J36). As the name suggests, it can be constructed by elongating a "triangular gyrobicupola," or cuboctahedron, by inserting a hexagonal prism between its two halves, which are congruent triangular cupolae (J3). Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola (J35).

Gyroelongated pentagonal cupolarotunda

In geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids (J47). As the name suggests, it can be constructed by gyroelongating a pentagonal cupolarotunda by inserting a decagonal antiprism between its two halves.

References

  1. ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics , 18: 169–200, doi:10.4153/cjm-1966-021-8, MR   0185507, Zbl   0132.14603 .
  2. ↑ Stephen Wolfram, "Gyroelongated triangular bicupola" from Wolfram Alpha. Retrieved July 30, 2010.
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