Gyroelongated pentagonal pyramid

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Short description: 11th Johnson solid (16 faces)
Gyroelongated pentagonal pyramid
TypeJohnson
J10 – J11 – J12
Faces15 triangles
1 pentagon
Edges25
Vertices11
Vertex configuration5(33.5)
1+5(35)
Symmetry groupC5v
Propertiescomposite, convex
Net

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.[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 J11.[4] It is an example of composite polyhedron.[5]

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:[2] A=153+5(5+25)4a2≈8.215a2,V=25+9524a3≈1.880a3.

It has the same three-dimensional symmetry group as the pentagonal pyramid: the cyclic group C5v 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 k-connected graph is one that remains connected whenever 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

References

  1. ↑ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, pp. 84–89, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4, https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84 .
  2. ↑ 2.0 2.1 Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8 .
  3. ↑ Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, Springer-Verlag, 2000, p. 457, ISBN 9780387986500, https://books.google.com/books?id=EJCSL9S6la0C&pg=PA457 .
  4. ↑ Uehara, Ryuhei (2020), Introduction to Computational Origami: The World of New Computational Geometry, Springer, p. 62, doi:10.1007/978-981-15-4470-5, ISBN 978-981-15-4470-5, https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 .
  5. ↑ Timofeenko, A. V. (2009), "Convex Polyhedra with Parquet Faces", Doklady Mathematics 80 (2): 720–723, doi:10.1134/S1064562409050238, https://www.interocitors.com/tmp/papers/timo-parquet.pdf .
  6. ↑ 6.0 6.1 Cheng, Peng (2023), Lanthanides: Fundamentals and Applications, Elsevier, p. 166, ISBN 978-0-12-822250-8, https://books.google.com/books?id=yousEAAAQBAJ&pg=PA166 .
  7. ↑ "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, 1966, doi:10.4153/cjm-1966-021-8 ; see table III, line 11.
  8. ↑ Kepert, David L. (1982), "Polyhedra", Inorganic Chemistry Concepts, 6, Springer, p. 14, doi:10.1007/978-3-642-68046-5_2, ISBN 978-3-642-68048-9, https://books.google.com/books?id=4QvpCAAAQBAJ&pg=PA14 .



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