Pentagonal orthocupolarotunda

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Pentagonal orthocupolarotunda
Pentagonal orthocupolarotunda.png
Type Johnson
J31 – J32 – J33
Faces 3×5 triangles
5 squares
2+5 pentagons
Edges 50
Vertices 25
Vertex configuration 10(3.4.3.5)
5(3.4.5.4)
2.5(3.5.3.5)
Symmetry group C5v
Dual polyhedron -
Properties convex
Net
Johnson solid 32 net.png

In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (J32). As the name suggests, it can be constructed by joining a pentagonal cupola (J5) and a pentagonal rotunda (J6) along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda (J33).

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 gyrobirotunda 43rd Johnson solid

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

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

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Elongated pentagonal orthobicupola 38th Johnson solid

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Elongated pentagonal gyrobicupola 39th Johnson solid

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Gyroelongated triangular cupola

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Pentagonal gyrocupolarotunda 33rd Johnson solid; pentagonal cupola and rotunda joined base-to-base

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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 orthocupolarotunda" from Wolfram Alpha. Retrieved July 24, 2010.
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