Editing Gyroelongated square pyramid

{{Short description|10th Johnson solid (13 faces)}}
{{Infobox polyhedron
 | image = gyroelongated_square_pyramid.png
 | type = [[Johnson solid|Johnson]]<br>{{math|[[elongated pentagonal pyramid|''J''{{sub|9}}]] – '''''J''{{sub|10}}''' – [[gyroelongated pentagonal pyramid|''J''{{sub|11}}]]}}
 | faces = 12 [[triangle]]s <br> 1 [[Square (geometry)|square]]
 | edges = 20
 | vertices = 9
 | symmetry = <math> C_{4v} </math>
 | vertex_config = <math> 1 \times 3^4 + 4 \times 3^3 \times 4 + 4 \times 3^5 </math>
 | properties = [[convex set|convex]], [[composite polyhedron|composite]]
 | net = Johnson solid 10 net.png
}}

In [[geometry]], the '''gyroelongated square pyramid''' is the [[Johnson solid]] that can be constructed by attaching an [[equilateral square pyramid]] to a [[square antiprism]]. It occurs in chemistry; for example, the [[capped square antiprismatic molecular geometry]].

== Construction ==
The gyroelongated square pyramid is [[Composite polyhedron|composite]], since it can constructed by attaching one [[equilateral square pyramid]] to the [[square antiprism]], a process known as the [[gyroelongation]].{{r|timofeenko-2010|rajwade}} This construction involves the covering of one of two square faces and replacing them with the four [[equilateral triangles]], so that the resulting polyhedron has twelve equilateral triangles and one square.{{r|berman}} The convex polyhedron in which all of the faces are regular is the [[Johnson solid]], and the gyroelongated square pyramid is one of them, enumerated as <math> J_{10} </math>, the tenth Johnson solid.{{r|uehara}}

== Properties ==
The surface area of a gyroelongated square pyramid with edge length <math> a </math> is:{{r|berman}}
<math display="block"> \left(1 + 3\sqrt{3}\right)a^2 \approx 6.196a^2, </math>
the area of twelve equilateral triangles and a square. Its volume:{{r|berman}}
<math display="block"> \frac{\sqrt{2} + 2\sqrt{4 + 3\sqrt{2}}}{6}a^3 \approx 1.193a^3, </math>
can be obtained by slicing the square pyramid and the square antiprism, after which adding their volumes.{{r|berman}}

It has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the square pyramid, the [[cyclic group]] <math> C_{4v} </math> of order eight. Its [[dihedral angle]] can be derived by calculating the angle of a square pyramid and square antiprism in the following:{{r|johnson}}
* the dihedral angle of an equilateral square pyramid between two adjacent triangles, approximately <math> 109.47^\circ </math>
* the dihedral angle of a square antiprism between two adjacent triangles, approximately <math> 127.55^\circ </math>, and between a triangle to its base is <math> 103.83^\circ </math>
* the dihedral angle between two adjacent triangles, on the edge where an equilateral square pyramid is attached to a square antiprism, is <math> 158.57^\circ</math>, for which by adding the dihedral angle of an equilateral square pyramid between its base and its lateral face <math> 54.74^\circ </math> and the dihedral angle of a square antiprism between two adjacent triangles.

== Applications ==
In [[stereochemistry]], the [[capped square antiprismatic molecular geometry]] can be described as the [[atom cluster]] of the gyroelongated square pyramid. An example is {{chem|[LaCl(H|2|O)|7|]|2|4+}}, a [[lanthanum]](III) complex with a La–La bond.{{r|greenwood-earnshaw}}

== References ==
{{Reflist|refs=

<ref name="berman">{{cite journal
 | 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="greenwood-earnshaw">{{cite book
 | last1 = Greenwood | first1 = Norman N. | author-link1 = Norman Greenwood
 | last2 = Earnshaw | first2 = Alan
 | year = 1997
 | title = Chemistry of the Elements
 | edition = 2nd
 | page = 917
 | publisher = [[Butterworth-Heinemann]]
 | isbn = 978-0-08-037941-8
}}</ref>

<ref name="johnson">{{cite journal
 | last = Johnson | first = Norman W. | author-link = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603| doi-access = free
}}</ref>

<ref name="rajwade">{{cite book
 | 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
 | publisher = Hindustan Book Agency
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}</ref>

<ref name="timofeenko-2010">{{cite journal
 | last = Timofeenko | first = A. V.
 | year = 2010
 | title = Junction of Non-composite Polyhedra
 | journal = St. Petersburg Mathematical Journal
 | volume = 21 | issue = 3 | pages = 483–512
 | doi = 10.1090/S1061-0022-10-01105-2
 | url = https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf
}}</ref>

<ref name="uehara">{{cite book
 | 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>

}}
== See also ==
* [[Gyroelongated square bipyramid]]

==External links==
* {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname =GyroelongatedSquarePyramid| title =Gyroelongated square pyramid}}

{{Johnson solids navigator}}

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