Editing Gyroelongated pentagonal pyramid

{{Short description|11th Johnson solid (16 faces)}}
{{Infobox polyhedron
 | image = Blue gyroelongated pentagonal pyramid.svg
 | type = [[Johnson solid|Johnson]]<br>{{math|[[gyroelongated square pyramid|''J''{{sub|10}}]] – '''''J''{{sub|11}}''' – [[triangular bipyramid|''J''{{sub|12}}]]}}
 | faces = 15 [[triangle]]s<br>1 [[pentagon]]
 | edges = 25
 | vertices = 11
 | symmetry = <math> C_{5 \mathrm{v}} </math>
 | vertex_config = {{math|5(3{{sup|3}}.5)<br>1+5(3{{sup|5}})}}
 | properties = [[composite polyhedron|composite]], [[convex set|convex]]
 | net = Gyroelongated pentagonal pyramid net.png
}}
[[File:J11 gyroelongated pentagonal pyramid.stl|thumb|3D model of a gyroelongated pentagonal pyramid]]

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.{{r|rajwade}} This pyramid covers the pentagonal faces, so the resulting polyhedron has 15 [[equilateral triangle]]s 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 (geometry)|diminishment]]; for this reason, it is also called the ''diminished icosahedron''.{{r|hartshorne}} Because the resulting polyhedron has the property of [[Convex set|convexity]] and its faces are [[regular polygon]]s, the gyroelongated pentagonal pyramid is a [[Johnson solid]], enumerated as the 11th Johnson solid <math> J_{11} </math>.{{r|uehara}} It is an example of [[composite polyhedron]].{{r|timofeenko-2009}}

== Properties ==
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:{{r|berman}}
<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 [[Point groups in three dimensions|three-dimensional symmetry group]] as the pentagonal pyramid: the cyclic group <math> C_{5 \mathrm{v}} </math> 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 (topology)|skeleton]] of a gyroelongated pentagonal pyramid can be represented in a [[planar graph]] with a [[k-vertex-connected graph|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 graph|factor-critical]].

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

== See also ==
* [[Metabidiminished icosahedron]]
* [[Tridiminished icosahedron]]

== References ==
{{reflist|refs=

<ref name="berman">{{citation
 | last = Berman | first = Martin
 | year = 1971
 | title = Regular-faced convex polyhedra
 | journal = Journal of the Franklin Institute
 | volume = 291
 | issue = 5
 | pages = 329–352
 | doi = 10.1016/0016-0032(71)90071-8
 | mr = 290245
}}.</ref>

<ref name="cheng">{{citation
 | last = Cheng | first = Peng
 | year = 2023
 | title = Lanthanides: Fundamentals and Applications
 | url = https://books.google.com/books?id=yousEAAAQBAJ&pg=PA166
 | page = 166
 | publisher = Elsevier
 | isbn = 978-0-12-822250-8
}}.</ref>

<ref name="hartshorne">{{citation
 | last = Hartshorne | first = Robin | author-link = Robin Hartshorne
 | year = 2000
 | title = Geometry: Euclid and Beyond
 | series = Undergraduate Texts in Mathematics
 | publisher = Springer-Verlag
 | isbn = 9780387986500
 | url = https://books.google.com/books?id=EJCSL9S6la0C&pg=PA457
 | page = 457
}}.</ref>

<ref name="johnson">{{citation
 | last = Johnson | first = Norman W. | author-link = Norman Johnson (mathematician)
 | doi = 10.4153/cjm-1966-021-8
 | journal = [[Canadian Journal of Mathematics]]
 | mr = 0185507
 | pages = 169–200
 | title = Convex polyhedra with regular faces
 | volume = 18
 | year = 1966
 | zbl = 0132.14603
}}; see table III, line 11.</ref>

<ref name="kepert">{{citation
 | last = Kepert | first = David L.
 | contribution = Polyhedra
 | url = https://books.google.com/books?id=4QvpCAAAQBAJ&pg=PA14
 | doi = 10.1007/978-3-642-68046-5_2
 | page = 14
 | publisher = Springer
 | title = Inorganic Chemistry Concepts
 | year = 1982| volume = 6
 | isbn = 978-3-642-68048-9
}}.</ref>

<ref name="rajwade">{{citation
 | last = Rajwade | first = A. R.
 | title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
 | series = Texts and Readings in Mathematics
 | year = 2001
 | url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
 | pages = 84&ndash;89
 | publisher = Hindustan Book Agency
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}.</ref>

<ref name="timofeenko-2009">{{citation
 | last = Timofeenko | first = A. V.
 | year = 2009
 | title = Convex Polyhedra with Parquet Faces
 | journal = Docklady Mathematics
 | url = https://www.interocitors.com/tmp/papers/timo-parquet.pdf
 | volume = 80 | issue = 2
 | pages = 720–723
 | doi = 10.1134/S1064562409050238
}}.</ref>

<ref name="uehara">{{citation
 | last = Uehara | first = Ryuhei
 | year = 2020
 | title = Introduction to Computational Origami: The World of New Computational Geometry
 | url = https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62
 | page = 62
 | publisher = Springer
 | isbn = 978-981-15-4470-5
 | doi = 10.1007/978-981-15-4470-5
 | s2cid = 220150682
}}.</ref>

}}

==External links==
* {{mathworld | urlname =GyroelongatedPentagonalPyramid| title =Gyroelongated pentagonal pyramid}}

{{Johnson solids navigator}}

[[Category:Composite polyhedron]]
[[Category:Johnson solids]]
[[Category:Pyramids (geometry)]]