Square gyrobicupola

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Square gyrobicupola
Square gyrobicupola.png
Type Bicupola,
Johnson
J28 – J29 – J30
Faces 8 triangles
2+8 squares
Edges 32
Vertices 16
Vertex configuration 8(3.4.3.4)
8(3.43)
Symmetry group D4d
Dual polyhedron Elongated square trapezohedron
Properties convex
Net
Johnson solid 29 net.png

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.

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 square gyrobicupola is the second in an infinite set of gyrobicupolae.

Related to the square gyrobicupola is the elongated square gyrobicupola. This polyhedron is created when an octagonal prism is inserted between the two halves of the square gyrobicupola. It is argued whether or not the elongated square gyrobicupola is an Archimedean solid because, although it meets every other standard necessary to be an Archimedean solid, it is not highly symmetric.

Formulae

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

The square gyrobicupola forms space-filling honeycombs with tetrahedra, cubes and cuboctahedra; and with tetrahedra, square pyramids, and elongated square bipyramids. (The latter unit can be decomposed into elongated square pyramids, cubes, and/or square pyramids). [3]

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Elongated square cupola 19th Johnson solid

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Elongated square gyrobicupola 37th Johnson solid

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Square orthobicupola 28th Johnson solid; 2 square cupolae joined base-to-base

In geometry, the square orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola.

Elongated pentagonal gyrobirotunda 43rd Johnson solid

In geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron, by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda.

Elongated pentagonal orthobirotunda 42nd Johnson solid

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids. Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda.

Elongated triangular pyramid 7th Johnson solid (7 faces)

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Elongated square pyramid 8th Johnson solid (9 faces)

In geometry, the elongated square pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a square pyramid by attaching a cube to its square base. Like any elongated pyramid, it is topologically self-dual.

Elongated triangular bipyramid 14th Johnson solid; triangular prism capped with tetrahedra

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

Elongated square bipyramid 15th Johnson solid; cube capped by 2 square pyramids

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Gyrobifastigium 26th Johnson solid (8 faces)

In geometry, the gyrobifastigium is the 26th Johnson solid. 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.

Pentagonal gyrobicupola 31st Johnson solid; 2 pentagonal cupolae joined base-to-base

In geometry, the pentagonal gyrobicupola is one of the Johnson solids. Like the pentagonal orthobicupola, it can be obtained by joining two pentagonal cupolae along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

Elongated pentagonal orthobicupola 38th Johnson solid

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Elongated pentagonal gyrobicupola 39th Johnson solid

In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola.

Elongated triangular cupola 18th Johnson solid

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

Triangular orthobicupola 27th Johnson solid; 2 triangular cupolae joined base-to-base

In geometry, the triangular orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by attaching two triangular cupolas 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 orthobicupola Johnson solid with 20 faces

In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids. As the name suggests, it can be constructed by elongating a triangular orthobicupola by inserting a hexagonal prism between its two halves. The resulting solid is superficially similar to the rhombicuboctahedron, with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry.

Elongated triangular gyrobicupola 36th Johnson solid

In geometry, the elongated triangular gyrobicupola is one of the Johnson solids. 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. Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola.

Elongated pentagonal gyrocupolarotunda 41st Johnson solid

In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola or the pentagonal rotunda through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda.

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, "Triangular gyrobicupola" from Wolfram Alpha. Retrieved July 23, 2010.
  3. ↑ "J29 honeycomb".
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