View source for Elongated square gyrobicupola

{{Short description|37th Johnson solid}}
{{Infobox polyhedron
 | image = elongated square gyrobicupola.png
 | type = [[Canonical polyhedron|Canonical]],<br>[[Johnson solid|Johnson]]<br>{{math|[[elongated triangular gyrobicupola|''J''{{sub|36}}]] – '''''J''{{sub|37}}''' – [[elongated pentagonal orthobicupola|''J''{{sub|38}}]]}}
 | faces = 8 [[triangle]]s<br>18 [[Square (geometry)|squares]]
 | edges = 48
 | vertices = 24
 | symmetry = <math> D_{4\mathrm{d}} </math>
 | vertex_config = <math> 8  + 16 (3 \cdot 4^3) </math>
 | properties = [[convex set|convex]],<br>singular [[vertex figure]]
 | net = Johnson solid 37 net.png
}}

In [[geometry]], the '''elongated square gyrobicupola''' is a polyhedron constructed by two [[square cupola]]s attaching onto the bases of [[octagonal prism]], with one of them rotated. It was once mistakenly considered a [[rhombicuboctahedron]] by many mathematicians. It is not considered to be an [[Archimedean solid]] because it lacks a set of global [[symmetries]] that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a [[Midsphere#Canonical polyhedron|canonical polyhedron]]. For this reason, it is also known as '''pseudo-rhombicuboctahedron''', '''Miller solid''',{{r|cromwell}} or '''Miller&ndash;Askinuze solid'''.{{r|johnson}}

== Construction ==
The elongated square gyrobicupola can be constructed similarly to the [[rhombicuboctahedron]], by attaching two regular [[square cupola]]s onto the bases of an [[octagonal prism]], a process known as [[Elongation (geometry)|elongation]]. The difference between these two polyhedrons is that one of the two square cupolas is twisted by 45&nbsp;degrees, a process known as ''gyration'', making the triangular faces staggered vertically.{{r|berman|cromwell}} The resulting polyhedron has 8 [[equilateral triangles]] and 18 [[square]]s.{{r|berman}} A [[Convex set|convex]] polyhedron in which all of the faces are [[regular polygon]]s is a [[Johnson solid]], and the elongated square gyrobicupola is among them, enumerated as the 37th Johnson solid <math> J_{37} </math>.{{r|francis}}
{{multiple image
 | image1 = Small rhombicuboctahedron.png
 | image2 = Exploded rhombicuboctahedron.png
 | image3 = Pseudorhombicuboctahedron.png
 | footer = Process of the construction of the elongated square gyrobicupola
 | total_width = 400
 | align = center
}}

The elongated square gyrobicupola may have been discovered by [[Johannes Kepler]] in his enumeration of the Archimedean solids, but its first clear appearance in print appears to be the work of [[Duncan Sommerville]] in 1905.{{r|sommerville}} It was independently rediscovered by [[J. C. P. Miller]] in 1930 by mistake while attempting to construct a model of the [[rhombicuboctahedron]]. This solid was discovered again by V. G. Ashkinuse in 1957.{{r|cromwell|ball|grunbaum}}

== Properties ==
An elongated square gyrobicupola with edge length <math> a </math> has a surface area:{{r|berman}}
<math display="block"> \left(18+2\sqrt{3}\right)a^2 \approx 21.464a^2, </math>
by adding the area of 8 equilateral triangles and 10 squares. Its volume can be calculated by slicing it into two square cupolas and one octagonal prism:{{r|berman}}
<math display="block"> \frac{12+10\sqrt{2}}{3}a^3 \approx 8.714a^3. </math>

[[File:J37 elongated square gyrobicupola.stl|thumb|3D model of an elongated square gyrobicupola]]
The elongated square gyrobicupola possesses [[Point groups in three dimensions|three-dimensional symmetry group]] <math> D_{4\mathrm{d}} </math> of order 16.  It is locally vertex-regular – the arrangement of the four faces incident on any vertex is the same for all vertices; this is unique among the Johnson solids. However, the manner in which it is "twisted" gives it a distinct "equator" and two distinct "poles", which in turn divides its vertices into 8 "polar" vertices (4 per pole) and 16 "equatorial" vertices. It is therefore not [[vertex-transitive]], and consequently not usually considered to be the 14th [[Archimedean solid]].{{r|cromwell|grunbaum|lz}}

The [[dihedral angle]] of an elongated square gyrobicupola can be ascertained in a similar way as the rhombicuboctahedron, by adding the dihedral angle of a square cupola and an octagonal prism:{{r|johnson}}
* the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom is that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a [[regular octagon]] 135°. The dihedral angle between two adjacent squares on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola square-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 45° + 90° = 135°. Therefore, the dihedral angle of a rhombicuboctahedron for every two adjacent squares is 135°.
* the dihedral angle of a rhombicuboctahedron square-to-triangle is that of a square cupola between those, 144.7°. The dihedral angle between square-to-triangle, on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola triangle-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 54.7° + 90° = 144.7°. Therefore, the dihedral angle of a rhombicuboctahedron for every square-to-triangle is 144.7°.

==Related polyhedra and honeycombs==
The elongated square gyrobicupola can form a space-filling [[Honeycomb (geometry)|honeycomb]] with the regular [[tetrahedron]], cube, and [[cuboctahedron]]. It can also form another honeycomb with the tetrahedron, [[square pyramid]] and various combinations of cubes, [[elongated square pyramid]]s, and [[elongated square bipyramid]]s.<ref>{{citation|url=http://woodenpolyhedra.web.fc2.com/J37.html|title=J37 honeycombs|access-date=2016-03-21|work=Gallery of Wooden Polyhedra}}</ref><br>

[[File:Pseudo-great rhombicuboctahedron.png|120px|thumb|right|The [[pseudo great rhombicuboctahedron]]]]
The [[pseudo great rhombicuboctahedron]] is a nonconvex analog of the pseudo-rhombicuboctahedron, constructed in a similar way from the [[nonconvex great rhombicuboctahedron]].

==In chemistry==
The [[vanadate|polyvanadate]] ion <nowiki>[</nowiki>[[vanadium|V]]<sub>18</sub>[[oxygen|O]]<sub>42</sub>]<sup>12−</sup> has a pseudo-rhombicuboctahedral structure, where each square face acts as the base of a VO<sub>5</sub> pyramid.<ref>{{Greenwood&Earnshaw2nd|page=986}}</ref>

== References ==
{{reflist|refs=
<ref name="ball">{{citation
 | last = Ball | first = Rouse
 | title = Mathematical recreations and essays
 | page = 137
 | edition = 11
 | year = 1939
 | editor-last = Coxeter | editor-first = H. S. M.
}}.</ref>

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

<ref name="cromwell">{{citation
 | last = Cromwell | first = Peter R.
 | title = Polyhedra
 | year = 1997
 | url = https://books.google.com/books?id=OJowej1QWpoC&pg=PA91
 | page = 91
 | publisher = [[Cambridge University Press]]
 | isbn = 978-0-521-55432-9
}}.</ref>

<ref name="francis">{{citation
 | last = Francis | first = Darryl
 | title = Johnson solids & their acronyms
 | journal = Word Ways
 | date = August 2013
 | volume = 46 | issue = 3 | page = 177
 | url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118
}}.</ref>

<ref name="grunbaum">{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.4171/EM/120
 | issue = 3
 | journal = [[Elemente der Mathematik]]
 | mr = 2520469
 | pages = 89–101
 | title = An enduring error
 | url = https://digital.lib.washington.edu/dspace/bitstream/handle/1773/4592/An_enduring_error.pdf
 | volume = 64
 | year = 2009| doi-access = free
 }} Reprinted in {{cite book|title=The Best Writing on Mathematics 2010|editor-first=Mircea|editor-last=Pitici|publisher=Princeton University Press|year=2011|pages=18–31}}.</ref>

<ref name="johnson">{{citation
 | last = Johnson | first = Norman W. | authorlink = 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="lz">{{citation
 | last1 = Lando | first1 = Sergei K.
 | last2 = Zvonkin | first2 = Alexander K.
 | year = 2004
 | title = Graphs on Surfaces and Their Applications
 | url = https://books.google.com/books?id=nFnyCAAAQBAJ&pg=PA114
 | page = 114
 | publisher = Springer
 | doi =10.1007/978-3-540-38361-1
 | isbn = 978-3-540-38361-1
}}.</ref>

<ref name="sommerville">{{citation
 | last = Sommerville | first = D. M. Y. | author-link = Duncan Sommerville
 | title = Semi-regular networks of the plane in absolute geometry
 | journal = Transactions of the Royal Society of Edinburgh
 | volume = 41
 | year = 1905
 | pages = 725–747
 | doi = 10.1017/s0080456800035560
 | url = https://zenodo.org/record/1428682
}}. As cited by {{harvtxt|Grünbaum|2009}}.</ref>

}}

==Further reading==
* {{citation | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }} Chapter 2: Archimedean polyhedra, prisma and antiprisms, p. 25 Pseudo-rhombicuboctahedron

==External links==
* {{Mathworld2 | urlname = ElongatedSquareGyrobicupola  | title = Elongated square gyrobicupola  | urlname2 = JohnsonSolid  | title2 = Johnson solid }}
* [http://www.georgehart.com/virtual-polyhedra/pseudo-rhombicuboctahedra.html George Hart: pseudo-rhombicuboctahedra]
{{Johnson solids navigator}}
[[Category:Johnson solids]]
[[Category:Pseudo-uniform polyhedra]]