Editing Disphenocingulum

{{short description|90th Johnson solid (22 faces)}}
{{Infobox polyhedron
 |image=Disphenocingulum.png
 |type=[[Johnson solid|Johnson]]<br>{{math|[[hebesphenomegacorona|''J''{{sub|89}}]] – '''''J''{{sub|90}}''' – [[bilunabirotunda|''J''{{sub|91}}]]}}
 |faces=20 [[triangle]]s<br>4 [[Square (geometry)|square]]s
 |edges=38
 |vertices=16
 |symmetry={{math|D{{sub|2d}}}}
 |vertex_config={{math|4(3{{sup|2}}.4{{sup|2}}) <br> 4(3{{sup|5}}) <br> 8(3{{sup|4}}.4)}}
 |properties=[[convex set|convex]], [[Elementary polyhedron|elementary]]
 |net=Johnson solid 90 net.png
}}
[[File:J90 disphenocingulum.stl|thumb|3D model of a disphenocingulum]]

In [[geometry]], the '''disphenocingulum''' is a [[Johnson solid]] with 20 equilateral triangles and 4 squares as its faces.

== Properties ==
The disphenocingulum is named by {{harvtxt|Johnson|1966}}. The prefix ''dispheno-'' refers to two wedgelike complexes, each formed by two adjacent lunes&mdash;a figure of two [[equilateral triangle]]s at the opposite sides of a [[square]]. The suffix ''-cingulum'', literally 'belt', refers to a band of 12 triangles joining the two wedges.{{r|johnson}} The resulting polyhedron has 20 equilateral triangles and 4 squares, making 24 faces.{{r|berman}}. All of the faces are [[Regular polygon|regular]], categorizing the disphenocingulum as a [[Johnson solid]]&mdash;a [[Convex set|convex]] polyhedron in which all of its faces are regular polygon&mdash;enumerated as 90th Johnson solid <math> J_{90} </math>.{{r|francis}}. It is an [[elementary polyhedron]], meaning that it cannot be separated by a plane into two small regular-faced polyhedra.{{r|cromwell}}

The surface area of a disphenocingulum with edge length <math> a </math> can be determined by adding all of its faces, the area of 20 equilateral triangles and 4 squares <math> (4 + 5\sqrt{3})a^2 \approx 12.6603a^2 </math>, and its volume is <math> 3.7776a^3 </math>.{{r|berman}}

== Cartesian coordinates ==
Let <math> a \approx 0.76713 </math> be the second smallest positive root of the [[polynomial]]
<math display="block"> \begin{align} &256x^{12} - 512x^{11} - 1664x^{10} + 3712x^9 + 1552x^8 - 6592x^7 \\ &\quad{} + 1248x^6 + 4352x^5 - 2024x^4 - 944x^3 + 672x^2 - 24x - 23 \end{align}</math>
and <math>h = \sqrt{2+8a-8a^2}</math> and <math>c = \sqrt{1-a^2}</math>. Then, the [[Cartesian coordinates]] of a disphenocingulum with edge length 2 are given by the union of the orbits of the points
<math display="block">\left(1,2a,\frac{h}{2}\right),\ \left(1,0,2c+\frac{h}{2}\right),\ \left(1+\frac{\sqrt{3-4a^2}}{c},0,2c-\frac{1}{c}+\frac{h}{2}\right)</math>
under the action of the [[Symmetry group|group]] generated by reflections about the xz-plane and the yz-plane.{{r|timofeenko}}

== References ==
{{reflist|refs=

<ref name="berman">{{cite journal
 | last = Berman | first = M.
 | 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">{{cite book
 | last = Cromwell | first = P. R.
 | title = Polyhedra
 | year = 1997
 | url = https://archive.org/details/polyhedra0000crom/page/87/mode/1up
 | publisher = [[Cambridge University Press]]
 | isbn = 978-0-521-66405-9
 | page = 86&ndash;87, 89
}}</ref>

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

<ref name="timofeenko">{{cite journal
 | last = Timofeenko | first = A. V.
 | year = 2009
 | title = The non-Platonic and non-Archimedean noncomposite polyhedra
 | journal = Journal of Mathematical Science
 | volume = 162 | issue = 5 | pages = 717
 | doi = 10.1007/s10958-009-9655-0
 | s2cid = 120114341
}}</ref>

}}

==External links==
* {{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname =Disphenocingulum| title = Disphenocingulum}}

{{Johnson solids navigator}}

[[Category:Elementary polyhedron]]
[[Category:Johnson solids]]