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	convex
Net

Last updated July 29, 2024

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]

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. 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°.^[5]

Related Research Articles

<span class="mw-page-title-main">Regular icosahedron</span> Polyhedron with 20 regular triangular faces

In geometry, the regular icosahedron is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.

In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

In geometry, the pentagonal bipyramid is a polyhedron with 10 triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an example of deltahedra, and of Johnson solid.

<span class="mw-page-title-main">Gyroelongated square pyramid</span> 10th Johnson solid (13 faces)

In geometry, the gyroelongated square pyramid is the Johnson solid that can be constructed by attaching an equilateral square pyramid to a square antiprism. It occurs in chemistry; for example, the square antiprismatic molecular geometry.

In geometry, pentagonal pyramid is a pyramid with a pentagon base and five triangular faces, having a total of six faces. It is categorized as Johnson solid if all of the edges are equal in length, forming equilateral triangular faces and a regular pentagonal base. The pentagonal pyramid can be found in many polyhedrons, including their construction. It also occurs in stereochemistry in pentagonal pyramidal molecular geometry.

In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. Many polyhedrons can be constructed involving the attachment of the base of a triangular cupola.

In geometry, the square cupola the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be used to construct many polyhedrons, particularly in other Johnson solids.

In geometry, the gyroelongated square bicupola is the Johnson solid constructed by attaching two square cupolae on each base of octagonal antiprism. It has the property of chirality.

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 square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid. It is topologically self-dual.

In geometry, the elongated triangular bipyramid or triakis triangular prism a polyhedron constructed from a triangular prism by attaching two tetrahedrons to its bases. It is an example of Johnson solid.

In geometry, the elongated square bipyramid is the polyhedron constructed by attaching two equilateral square pyramids onto a cube's faces that are opposite each other. It can also be seen as 4 lunes linked together with squares to squares and triangles to triangles. It is also been named the pencil cube or 12-faced pencil cube due to its shape.

In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.

In geometry, the augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto the square face of a triangular prism. As a result, it is an example of Johnson solid. It can be visualized as the chemical compound, known as capped trigonal prismatic molecular geometry.

In geometry, the biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of Johnson solid. It can be found in stereochemistry in bicapped trigonal prismatic molecular geometry.

In geometry, the augmented pentagonal prism is a polyhedron that can be constructed by attaching an equilateral square pyramid onto the square face of pentagonal prism. It is an example of Johnson solid.

In geometry, the biaugmented pentagonal prism is a polyhedron constructed from a pentagonal prism by attaching two equilateral square pyramids onto each of its square faces. It is an example of Johnson solid.

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

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 .
1 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, 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 .
↑ 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.

External links

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

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[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] 1 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 .

[johnson-5] 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.

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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
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

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	convex
Net