Gyroelongated pentagonal pyramid

Gyroelongated pentagonal pyramid
Gyroelongated pentagonal pyramid
Type	Johnson; J10 – J11 – J12
Faces	15 triangles; 1 pentagon
Edges	25
Vertices	11
Vertex configuration	5(33.5); 1+5(35)
Symmetry group
Properties	composite, convex
Net

In geometry, the gyroelongated pentagonal pyramid is a polyhedron constructed by attaching a pentagonal antiprism to the base of a pentagonal pyramid. An alternative name is diminished icosahedron because it can be constructed by removing a pentagonal pyramid from a regular icosahedron.

Construction

The gyroelongated pentagonal pyramid can be constructed from a pentagonal antiprism by attaching a pentagonal pyramid onto its pentagonal face.^[1] This pyramid covers the pentagonal faces, so the resulting polyhedron has 15 equilateral triangles and 1 regular pentagon as its faces.^[2] Another way to construct it is started from the regular icosahedron by cutting off one of two pentagonal pyramids, a process known as diminishment; for this reason, it is also called the diminished icosahedron.^[3] Because the resulting polyhedron has the property of convexity and its faces are regular polygons, the gyroelongated pentagonal pyramid is a Johnson solid, enumerated as the 11th Johnson solid $J_{11}$ .^[4] It is an example of composite polyhedron.^[5]

Properties

The surface area of a gyroelongated pentagonal pyramid $A$ can be obtained by summing the area of 15 equilateral triangles and 1 regular pentagon. Its volume $V$ can be ascertained either by slicing it off into both a pentagonal antiprism and a pentagonal pyramid, after which adding them up; or by subtracting the volume of a regular icosahedron to a pentagonal pyramid. With edge length $a$ , they are:^[2] ${\begin{aligned}A&={\frac {15{\sqrt {3}}+{\sqrt {5(5+2{\sqrt {5}})}}}{4}}a^{2}\approx 8.215a^{2},\\V&={\frac {25+9{\sqrt {5}}}{24}}a^{3}\approx 1.880a^{3}.\end{aligned}}$

It has the same three-dimensional symmetry group as the pentagonal pyramid: the cyclic group $C_{5\mathrm {v} }$ of order 10.^[6] Its dihedral angle can be obtained by involving the angle of a pentagonal antiprism and pentagonal pyramid: its dihedral angle between triangle-to-pentagon is the pentagonal antiprism's angle between that 100.8°, and its dihedral angle between triangle-to-triangle is the pentagonal pyramid's angle 138.2°.^[7]

According to Steinitz's theorem, the skeleton of a gyroelongated pentagonal pyramid can be represented in a planar graph with a 3-vertex connected. This graph is obtained by removing one of the icosahedral graph's vertices, an odd number of vertices of 11, resulting in a graph with a perfect matching. Hence, the graph is 2-vertex connected claw-free graph, an example of factor-critical.

Appearance

The gyroelongated pentagonal pyramid has appeared in stereochemistry, wherein the shape resembles the molecular geometry known as capped pentagonal antiprism.^[8]^[6]

References

^ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, pp. 84–89, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4.
^ ^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.
^ Hartshorne, Robin (2000), Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, Springer-Verlag, p. 457, ISBN 9780387986500.
^ Uehara, Ryuhei (2020), Introduction to Computational Origami: The World of New Computational Geometry, Springer, p. 62, doi:10.1007/978-981-15-4470-5, ISBN 978-981-15-4470-5, S2CID 220150682.
^ Timofeenko, A. V. (2009), "Convex Polyhedra with Parquet Faces" (PDF), Docklady Mathematics, 80 (2): 720–723, doi:10.1134/S1064562409050238.
^ ^a ^b Cheng, Peng (2023), Lanthanides: Fundamentals and Applications, Elsevier, p. 166, ISBN 978-0-12-822250-8.
^ 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; see table III, line 11.
^ Kepert, David L. (1982), "Polyhedra", Inorganic Chemistry Concepts, vol. 6, Springer, p. 14, doi:10.1007/978-3-642-68046-5_2, ISBN 978-3-642-68048-9.

External links

Weisstein, Eric W. "Gyroelongated pentagonal pyramid". MathWorld.

[rajwade-1] Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, pp. 84–89, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4.

[berman-2] 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.

[hartshorne-3] Hartshorne, Robin (2000), Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, Springer-Verlag, p. 457, ISBN 9780387986500.

[uehara-4] Uehara, Ryuhei (2020), Introduction to Computational Origami: The World of New Computational Geometry, Springer, p. 62, doi:10.1007/978-981-15-4470-5, ISBN 978-981-15-4470-5, S2CID 220150682.

[timofeenko-2009-5] Timofeenko, A. V. (2009), "Convex Polyhedra with Parquet Faces" (PDF), Docklady Mathematics, 80 (2): 720–723, doi:10.1134/S1064562409050238.

[cheng-6] Cheng, Peng (2023), Lanthanides: Fundamentals and Applications, Elsevier, p. 166, ISBN 978-0-12-822250-8.

[johnson-7] 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; see table III, line 11.

[kepert-8] Kepert, David L. (1982), "Polyhedra", Inorganic Chemistry Concepts, vol. 6, Springer, p. 14, doi:10.1007/978-3-642-68046-5_2, ISBN 978-3-642-68048-9.

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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)

Gyroelongated pentagonal pyramid

Gyroelongated pentagonal pyramid

Gyroelongated pentagonal pyramid

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Root Wikipedia Article

Gyroelongated pentagonal pyramid

Construction

Properties

Appearance

See also

References

External links

Gyroelongated pentagonal pyramid

Type	Johnson $J 10 - J 11 - J 12$
Faces	15 triangles 1 pentagon
Edges	25
Vertices	11
Vertex configuration	$5(3 3 .5) 1+5(3 5)$
Symmetry group	$C_{5\mathrm {v} }$
Properties	composite, convex
Net