Gyroelongated pentagonal pyramid
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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.
ConstructionEdit
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 <math> J_{11} </math>.Template:R It is an example of composite polyhedron.Template:R
PropertiesEdit
The surface area of a gyroelongated pentagonal pyramid <math> A </math> can be obtained by summing the area of 15 equilateral triangles and 1 regular pentagon. Its volume <math> V </math> 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 <math> a </math>, they are:Template:R <math display="block"> \begin{align}
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{align} </math>
It has the same three-dimensional symmetry group as the pentagonal pyramid: the cyclic group <math> C_{5 \mathrm{v}} </math> of order 10.Template:R 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
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.
AppearanceEdit
The gyroelongated pentagonal pyramid has appeared in stereochemistry, wherein the shape resembles the molecular geometry known as capped pentagonal antiprism.Template:R