View source for Pentagonal pyramid

{{Short description|Pyramid with a pentagon base}}
{{good article}}
{{Infobox polyhedron
 | image = [[file:Pentagonal pyramid.png|frameless]]
 | type = [[Pyramid (geometry)|Pyramid]]<br>[[Johnson solid|Johnson]]<br />{{math|[[square pyramid|''J''{{sub|1}}]] – '''''J''{{sub|2}}''' – [[triangular cupola|''J''{{sub|3}}]]}}
 | faces = 5 [[triangle]]s<br />1 [[pentagon]]
 | edges = 10
 | vertices = 6
 | symmetry = <math> C_{5\mathrm{v}} </math>
 | vertex_config = <math> 5 \times (3^2 \times 5) + 1 \times 3^5 </math>{{sfnp|Johnson|1966}}
 | angle = As a Johnson solid:<br>{{bullet list|triangle-to-triangle: 138.19°|triangle-to-pentagon: 37.37°}}
 | dual = [[self-dual]]
 | properties = [[convex set|convex]],<br>[[Elementary polyhedron|elementary]] (Johnson solid)
 | net = [[file:pentagonal pyramid flat.svg|frameless]]
}}
In [[geometry]], a '''pentagonal pyramid''' is a [[Pyramid (geometry)|pyramid]] with a [[pentagon]] base and five triangular faces, having a total of six faces. It is categorized as a [[Johnson solid]] if all of the edges are equal in length, forming [[Equilateral triangle|equilateral triangular]] faces and a [[regular pentagon]]al base.

Pentagonal pyramids occur as pieces and tools in the construction of many polyhedra. They also appear in the field of [[natural science]], as in [[stereochemistry]] where the shape can be described as the [[pentagonal pyramidal molecular geometry]], as well as the study of [[Capsid|shell]] assembling in the underlying [[potential energy surface]]s and [[disclination]] in [[fiveling]]s and related shapes such as pyramidal [[copper]] and other metal [[nanowires]].

== Properties ==
A pentagonal pyramid has six vertices, ten edges, and six faces. One of its faces is [[pentagon]], a ''base'' of the pyramid; five others are [[triangles]].<ref>{{multiref
|{{harvp|Ball|Coxeter|1987|p=[https://books.google.com/books?id=Ze5LDwAAQBAJ&pg=PA130 130]}}
|{{harvp|Grgić|Karakašić|Ivandić|Glavaš|2022|p=[https://books.google.com/books?id=IaxTEAAAQBAJ&pg=PA476 476]}}
}}</ref> Five of the edges make up the pentagon by connecting its five vertices, and the other five edges are known as the lateral edges of the pyramid, meeting at the sixth vertex called the [[Apex (geometry)|apex]].{{sfnp|Smith|2000|p=[https://books.google.com/books?id=B0khWEZmOlwC&pg=PA98 98]}} A pentagonal pyramid is said to be ''regular'' if its base is [[Circumscribed polygon|circumscribed]] in a circle that forms a [[regular pentagon]], and it is said to be ''right'' if its altitude is erected perpendicularly to the base's center.<ref>{{multiref
 |{{harvp|Calter|Calter|2011|p=[https://books.google.com/books?id=4fHwTZK3JEIC&pg=PA198 198]}}
 |{{harvp|Polya|1954|p=[https://books.google.com/books?id=-TWTcSa19jkC&pg=PA138 138]}}
}}</ref>

Like other right pyramids with a regular polygon as a base, this pyramid has [[pyramidal symmetry]] of [[cyclic group]] <math>C_{5\mathrm{v}}</math>: the pyramid is left invariant by rotations of one, two, three, four-fifths around its [[axis of symmetry]], the line connecting the apex to the center of the base. It is also [[mirror symmetric]] relative to any perpendicular plane passing through a bisector of the base.{{sfnp|Johnson|1966}} It can be represented as the [[wheel graph]] <math> W_5 </math>, meaning its [[Skeleton (topology)|skeleton]] can be interpreted as a pentagon in which its five vertices connects a vertex in the center called the [[universal vertex]].{{sfnp|Pisanski|Servatius|2013|p=[https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21 21]}} It is [[Self-dual polyhedron|self-dual]], meaning its [[dual polyhedron]] is the pentagonal pyramid itself.{{sfnp|Wohlleben|2019|p=[https://books.google.com/books?id=rEpjDwAAQBAJ&pg=PA485 485–486]}}

[[File:J2 pentagonal pyramid.stl|thumb|left|3D model of a pentagonal pyramid]]
When all edges are equal in length, the five triangular faces are [[Equilateral triangle|equilateral]] and the base is a regular pentagon. Because this pyramid remains [[Convex set|convex]] and all of its faces are [[regular polygon]]s, it is classified as the second [[Johnson solid]] <math> J_2 </math>.{{sfnp|Uehara|2020|p=[https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 62]}} The [[dihedral angle]] between two adjacent triangular faces is approximately 138.19° and that between the triangular face and the base is 37.37°.{{sfnp|Johnson|1966}} It is an [[elementary polyhedra|elementary polyhedron]], meaning that it cannot be separated by a plane to create two small convex polyhedrons with regular faces.<ref>{{multiref
|{{harvp|Hartshorne|2000|p=[https://books.google.com/books?id=EJCSL9S6la0C&pg=PA464 464]}}
|{{harvp|Johnson|1966}}
}}</ref> A [[polyhedron]]'s [[surface area]] is the sum of the areas of its faces. Therefore, the surface area of a pentagonal pyramid is the sum of the areas of the five triangles and the one pentagon. The volume of every pyramid equals one-third of the area of its base multiplied by its height. So, the volume of a pentagonal pyramid is one-third of the product of the height and a pentagonal pyramid's area.{{sfnp|Calter|Calter|2011|p=[https://books.google.com/books?id=4fHwTZK3JEIC&pg=PA198 198]}} In the case of Johnson solid with edge length <math> a </math>, its surface area <math> A </math> and volume <math>V</math> are:{{sfnp|Berman|1971}}
<math display="block"> \begin{align}
 A &= \frac{a^2}{2}\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)} \approx 3.88554a^2, \\
 V &= \frac{5 + \sqrt{5}}{24} a^3 \approx 0.30150a^3.
\end{align} </math>

== Applications ==
[[File:SmallStellatedDodecahedron.gif|thumb|upright=1|Pentagonal pyramids can be found in a small stellated dodecahedron]]
Pentagonal pyramids can be found as components of many polyhedrons.  Attaching its base to the pentagonal face of another polyhedron is an example of the construction process known as [[Augmentation (geometry)|augmentation]], and attaching it to [[Prism (geometry)|prisms]] or [[antiprism]]s is known as [[Elongation (geometry)|elongation]] or [[gyroelongation]], respectively.{{sfnp|Slobodan|Obradović|Ðukanović|2015}} Examples of polyhedrons are the [[pentakis dodecahedron]] is constructed from the [[dodecahedron]] by attaching the base of pentagonal pyramids onto each pentagonal face, [[small stellated dodecahedron]] is constructed from a [[regular dodecahedron]] [[Stellation|stellated]] by pentagonal pyramids, and a [[regular icosahedron]] constructed from a [[pentagonal antiprism]] by attaching two pentagonal pyramids onto its pentagonal bases.<ref>{{multiref
|{{harvp|Çolak|Gelişgen|2015}}
|{{harvp|Kappraff|2001|p=[https://books.google.com/books?id=twF7pOYXSTcC&pg=PA309 309]}}
|{{harvp|Silvester|2001|p=[https://books.google.com/books?id=VtH_QG6scSUC&pg=PA141 140&ndash;141]}}
}}</ref> Some Johnson solids are constructed by either augmenting pentagonal pyramids or augmenting other shapes with pentagonal pyramids: an [[elongated pentagonal pyramid]] <math> J_9 </math>, a [[gyroelongated pentagonal pyramid]] <math> J_{11} </math>, a [[pentagonal bipyramid]] <math> J_{13} </math>, an [[elongated pentagonal bipyramid]] <math> J_{16} </math>, an [[augmented dodecahedron]] <math> J_{58} </math>, a [[parabiaugmented dodecahedron]] <math> J_{59} </math>, a [[metabiaugmented dodecahedron]] <math> J_{60} </math>, and a [[triaugmented dodecahedron]] <math> J_{61} </math>.<ref>{{harvtxt|Rajwade|2001}}, pp. [https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84 84&ndash;88]. See Table 12.3, where <math> P_n </math> denotes the {{nowrap|1=<math>n</math>-sided}} prism and <math> A_n </math> denotes the {{nowrap|<math>n</math>-sided}} antiprism.</ref> Relatedly, the removal of a pentagonal pyramid from polyhedra is an example of a technique known as [[Diminishment (geometry)|diminishment]]; the [[metabidiminished icosahedron]] <math> J_{62} </math> and [[tridiminished icosahedron]] <math> J_{63} </math> are the examples in which their constructions begin by removing pentagonal pyramids from a regular icosahedron.{{sfnp|Gailiunas|2001}}

In [[stereochemistry]], an [[atom cluster]] can have a [[Pentagonal pyramidal molecular geometry|pentagonal pyramidal geometry]]. This molecule has a main-group element with one active [[lone pair]] of [[Electron|electrons]], which can be described by a model that predicts the geometry of molecules known as [[VSEPR theory]].{{sfnp|Petrucci|Harwood|Herring|2002|p=[https://books.google.com/books?id=EZEoAAAAYAAJ&pg=PA414 414]}} An example of a molecule with this structure is [[Boranes#Chemical formula and naming conventions|nido]]-cage [[carbonate]] CB<sub>5</sub>H<sub>9</sub>.{{sfnp|Macartney|2017|p=[https://books.google.com/books?id=BOlGDgAAQBAJ&pg=RA1-PA482 482]}}

{{harvtxt|Fejer|James|Hernández-Rojasc|Wales|2009}} modeled the formation of [[virus]] shells, known as [[Capsid|capsids]], from pieces shaped like pentagonal and hexagonal pyramids. These shapes were chosen to resemble those of the protein subunits of natural viruses. By appropriately choosing the attractive and repulsive forces between pyramids, they found that the pyramids could self-assemble into icosahedral shells reminiscent of those found in nature.{{sfnp|Fejer|James|Hernández-Rojasc|Wales|2009}}

{{harvtxt|Gryzunova|2017}} studied the [[Stress relaxation|relaxation]] of internal elastic [[Stress (mechanics)|stress]] fields due to [[disclination]]s in twinned [[copper]] particles. Such a shape is the pentagonal pyramid, which allows growth to a large size and preserves symmetry. This can be done by activating [[cathode]] by the process of initial crystal growth in the [[electrolyte]], by the movement of [[aluminum]] and [[silicon oxide]]s' abrasive particles.{{sfnp|Gryzunova|2017}}

== References ==
=== Notes ===
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{{refend}}

== External links ==
* {{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname = PentagonalPyramid | title = Pentagonal pyramid}}
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra ( [[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_pyramid_(J2).wrl model])

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