Gyroelongated pentagonal pyramid

Template:Short description Template:Infobox polyhedron File:J11 gyroelongated pentagonal pyramid.stl

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.Template:R This pyramid covers the pentagonal faces, so the resulting polyhedron has 15 equilateral triangles and 1 regular pentagon as its faces.Template:R 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.Template:R 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 $J_{11}$.Template:R

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:Template:R $$\begin{array}{cc}A& =\frac{15\sqrt{3}+\sqrt{5(5+2\sqrt{5})}}{4}{a}^2\approx 8.215a^2,\\ V& =\frac{25+9\sqrt{5}}{24}{a}^3\approx 1.880a^3.\end{array}$$

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°.Template:R

See also

• Metabidiminished icosahedron
• Tridiminished icosahedron

External links

• Template:Mathworld