Pentagonal orthobirotunda

Pentagonal orthobirotunda
Pentagonal orthobirotunda
Type	Birotunda,; Johnson; J33 – J34 – J35
Faces	2x10 triangles; 2+10 pentagons
Edges	60
Vertices	30
Vertex configuration	10(32.52); 2.10(3.5.3.5)
Symmetry group	D5h
Properties	convex
Net

In geometry, the pentagonal orthobirotunda is a polyhedron constructed by attaching two pentagonal rotundae along their decagonal faces, matching like faces. It is an example of Johnson solid.

Construction

The pentagonal orthobirotunda is constructed by attaching two pentagonal rotundas to their base, covering decagon faces. The resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to icosidodecahedron (or pentagonal gyrobirotunda), an Archimedean solid: the difference is one of its rotundas twisted around 36°, making the pentagonal faces connect to the triangular one, a process known as gyration.^[1]^[2] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The pentagonal orthobirotunda is one of them, enumerated as the 34th Johnson solid $J_{34}$ .^[3]

The difference between icosidodecahedron and pentagonal orthobirotunda, and its dissection.

Properties

The surface area of an icosidodecahedron $A$ can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron $V$ can be determined by slicing it off into two pentagonal rotunda, after which summing up their volumes. Therefore, its surface area and volume can be formulated as:^[1] ${\begin{aligned}A&=\left(5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}&\approx 29.306a^{2}\\V&={\frac {45+17{\sqrt {5}}}{6}}a^{3}&\approx 13.836a^{3}.\end{aligned}}$

References

^ ^a ^b Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
^ Ogievetsky, O.; Shlosman, S. (2021). "Platonic compounds and cylinders". In Novikov, S.; Krichever, I.; Ogievetsky, O.; Shlosman, S. (eds.). Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry. American Mathematical Society. p. 477. ISBN 978-1-4704-5592-7.
^ Francis, Darryl (2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.

External links

Weisstein, Eric W., "Pentagonal orthobirotunda" ("Johnson solid") at MathWorld.

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[berman-1] Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.

[os-2] Ogievetsky, O.; Shlosman, S. (2021). "Platonic compounds and cylinders". In Novikov, S.; Krichever, I.; Ogievetsky, O.; Shlosman, S. (eds.). Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry. American Mathematical Society. p. 477. ISBN 978-1-4704-5592-7.

[francis-3] Francis, Darryl (2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.

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v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Other elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)