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.

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	${\displaystyle C_{5\mathrm {v} }}$
Properties	composite, convex
Net

3D model of a gyroelongated pentagonal pyramid

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