Pentagonal gyrocupolarotunda

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Pentagonal gyrocupolarotunda
Pentagonal gyrocupolarotunda.png
Type Johnson
J32 – J33 – J34
Faces 3×5 triangles
5 squares
2+5 pentagons
Edges 50
Vertices 25
Vertex configuration 10(32.4.5)
5(3.4.5.4)
2.5(3.5.3.5)
Symmetry group C5v
Dual polyhedron -
Properties convex
Net
Johnson solid 33 net.png

In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids (J33). Like the pentagonal orthocupolarotunda (J32), it can be constructed by joining a pentagonal cupola (J5) and a pentagonal rotunda (J6) along their decagonal bases. The difference is that in this solid, the two halves are rotated 36 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]

Formulae

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

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Elongated pentagonal rotunda

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Elongated pentagonal gyrobirotunda 43rd Johnson solid

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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.

Pentagonal cupola 5th Johnson solid (12 faces)

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Elongated pentagonal cupola 20th Johnson solid

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Pentagonal orthobicupola 30th Johnson solid; 2 pentagonal cupolae joined base-to-base

In geometry, the pentagonal orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by joining two pentagonal cupolae along their decagonal bases, matching like faces. A 36-degree rotation of one cupola before the joining yields a pentagonal gyrobicupola.

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

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

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.

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.

Pentagonal orthocupolarotunda 32nd Johnson solid; pentagonal cupola and rotunda joined base-to-base

In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda.

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.

Elongated pentagonal orthocupolarotunda 40th Johnson solid

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

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, "Pentagonal gyrocupolarotunda" from Wolfram Alpha. Retrieved July 24, 2010.
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