Gyroelongated pentagonal pyramid | |
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Type | Johnson J10 – J11 – J12 |
Faces | 15 triangles 1 pentagon |
Edges | 25 |
Vertices | 11 |
Vertex configuration | 5(33.5) 1+5(35) |
Symmetry group | |
Properties | composite, convex |
Net | |
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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.
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 . [4] It is an example of composite polyhedron. [5]
The surface area of a gyroelongated pentagonal pyramid can be obtained by summing the area of 15 equilateral triangles and 1 regular pentagon. Its volume 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 , they are: [2]
It has the same three-dimensional symmetry group as the pentagonal pyramid: the cyclic group 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 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. [8] [6]