Gyroelongated pentagonal pyramid

Gyroelongated pentagonal pyramid
Type	Johnson J10 – J11 – J12
Faces	15 triangles 1 pentagon
Edges	25
Vertices	11
Vertex configuration	5(3^3.5) 1+5(3^5)
Symmetry group	${\displaystyle C_{5\mathrm {v} }}$
Properties	composite, convex
Net

3D model of a gyroelongated pentagonal pyramid

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 ${\displaystyle J_{11}}$.^[4] It is an example of composite polyhedron.^[5]

Properties

The surface area of a gyroelongated pentagonal pyramid ${\displaystyle A}$ can be obtained by summing the area of 15 equilateral triangles and 1 regular pentagon. Its volume ${\displaystyle V}$ 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 ${\displaystyle a}$, they are:^[2] $${\displaystyle {\begin{aligned}A&={\frac {15{\sqrt {3}}+{\sqrt {5(5+2{\sqrt {5}})}}}{4}}a^{2}\approx 8.215a^{2},\\V&={\frac {25+9{\sqrt {5}}}{24}}a^{3}\approx 1.880a^{3}.\end{aligned}}}$$

It has the same three-dimensional symmetry group as the pentagonal pyramid: the cyclic group ${\displaystyle C_{5\mathrm {v} }}$ 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 any convex polyhedron can be represented as a planar graph that is 3-vertex connected. A planar graph is one that can be drawn on a flat sheet with no edges crossing. A ${\displaystyle k}$-connected graph is one that remains connected whenever ${\displaystyle k-1}$ vertices are removed. This graph is obtained by removing one of the icosahedral graph's vertices, leaving 11 vertices, an odd number, resulting in a graph with a perfect matching. Hence, the graph is a 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]

See also

• Metabidiminished icosahedron
• Tridiminished icosahedron

References

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.
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.
3. Hartshorne, Robin (2000), Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, Springer-Verlag, p. 457, ISBN 9780387986500.
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.
5. Timofeenko, A. V. (2009), "Convex Polyhedra with Parquet Faces" (PDF), Doklady Mathematics, 80 (2): 720–723, doi:10.1134/S1064562409050238.
6. Cheng, Peng (2023), Lanthanides: Fundamentals and Applications, Elsevier, p. 166, ISBN 978-0-12-822250-8.
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.
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.

External links

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