Elongated pentagonal orthobirotunda

Elongated pentagonal orthobirotunda
Elongated pentagonal orthobirotunda
Type	Johnson ; J41 – J42 – J43
Faces	2.10 triangles ; 2.5 squares ; 2+10 pentagons
Edges	80
Vertices	40
Vertex configuration	20(3.42.5); 2.10(3.5.3.5)
Symmetry group	D5h
Dual polyhedron	-
Properties	convex
Net

Last updated June 11, 2022

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids ( $J 42$ ). Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda ( $J 34$ ) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae ( $J 6$ ) through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda ( $J 43$ ).

Formulae

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

V={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 21.5297...a^{3}

A=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\approx 39.306...a^{2}

Related Research Articles

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

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

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

In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J₂₁). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J₆) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J₄₂) with one pentagonal rotunda removed.

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.

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.

In geometry, the elongated triangular pyramid is one of the Johnson solids. 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.

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.

In geometry, the elongated pentagonal cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal cupola by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola with its "lid" removed.

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.

In geometry, the pentagonal orthobicupola is one of the Johnson solids (J₃₀). As the name suggests, it can be constructed by joining two pentagonal cupolae (J₅) along their decagonal bases, matching like faces. A 36-degree rotation of one cupola before the joining yields a pentagonal gyrobicupola (J₃₁).

In geometry, the pentagonal gyrobicupola is one of the Johnson solids (J₃₁). Like the pentagonal orthobicupola (J₃₀), it can be obtained by joining two pentagonal cupolae (J₅) along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids (J₃₈). As the name suggests, it can be constructed by elongating a pentagonal orthobicupola (J₃₀) 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 (J₃₉).

In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (J₃₉). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola (J₃₁) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae (J₅) through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola (J₃₈).

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.

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).

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.

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).

In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids (J₄₁). As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda (J₃₃) by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola (J₅) or the pentagonal rotunda (J₆) through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda (J₄₀).

In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids (J₄₀). As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda (J₃₂) 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 (J₄₁).

References

↑ 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 .
↑ Stephen Wolfram, "Elongated pentagonal orthobirotunda" from Wolfram Alpha. Retrieved July 26, 2010.

External links

Eric W. Weisstein, Elongated pentagonal orthobirotunda ( Johnson solid ) at MathWorld.

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[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, "Elongated pentagonal orthobirotunda" from Wolfram Alpha. Retrieved July 26, 2010.

[1]

[2]

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
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)

Elongated pentagonal orthobirotunda

Type	Johnson $J 41 - J 42 - J 43$
Faces	2.10 triangles 2.5 squares 2+10 pentagons
Edges	80
Vertices	40
Vertex configuration	$20(3.4 2 .5) 2.10(3.5.3.5)$
Symmetry group	$D 5h$
Dual polyhedron	-
Properties	convex
Net

Elongated pentagonal orthobirotunda

Contents

Formulae

Related Research Articles

References

External links