gyroelongated pentagonal pyramid

The gyroelongated pentagonal pyramid can be constructed from a pentagonal antiprism by attaching a pentagonal pyramid onto its pentagonal face.{{r|rajwade}} This pyramid covers the pentagonal faces, so the resulting polyhedron has 15 equilateral triangles and 1 regular pentagon as its faces.{{r|berman}} 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.{{r|hartshorne}} 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 $J\_\{11\}$.{{r|uehara}} It is an example of composite polyhedron.{{r|timofeenko-2009}}

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:{{r|berman}}

$$\backslash begin\{align\}$$

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}

It has the same three-dimensional symmetry group as the pentagonal pyramid: the cyclic group $C\_\{5\; \backslash mathrm\{v\}\}$ of order 10.{{r|cheng}} 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°.{{r|johnson}}

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.

The gyroelongated pentagonal pyramid has appeared in stereochemistry, wherein the shape resembles the molecular geometry known as capped pentagonal antiprism.{{r|kepert|cheng}}

• Metabidiminished icosahedron
• Tridiminished icosahedron

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• {{mathworld | urlname =GyroelongatedPentagonalPyramid| title =Gyroelongated pentagonal pyramid}}

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Category:Composite polyhedron

Category:Johnson solids

Category:Pyramids (geometry)