View source for Icositetrahedron

{{short description|Polyhedron with 24 faces}}
{{Multiple issues|{{more footnotes needed|date=June 2020}}{{more citations needed|date=June 2020}}}}

{| class=wikitable align=right
|- align=center
|[[File:Triakisoctahedron.jpg|200px]]<BR>[[Triakis octahedron]]
|[[File:Tetrakishexahedron.jpg|200px]]<BR>[[Tetrakis hexahedron]]
|- align=center
|[[File:Deltoidalicositetrahedron.jpg|200px]]<BR>[[Deltoidal icositetrahedron]]
|[[File:Pentagonalicositetrahedroncw.jpg|200px]]<BR>[[Pentagonal icositetrahedron]]
|}
In geometry, an '''icositetrahedron'''<ref>{{Cite web |title=Greek numerical prefixes |url=https://www.georgehart.com/virtual-polyhedra/greek-prefixes.html |access-date=2025-02-02 |website=www.georgehart.com}}</ref> refers to a polyhedron with 24 faces, none of which are [[Regular polyhedron|regular polyhedra]]. However, many are composed of regular polygons, such as the [[triaugmented dodecahedron]] and the [[disphenocingulum]]. Some icositetrahedra are near-[[Sphere|spherical]], but are not composed of regular polygons.  A minimum of 14 vertices is required to form a icositetahedron.<ref>{{Cite web |title=Enumeration of Polyhedra - Numericana |url=http://www.numericana.com/data/polycount.htm |access-date=2025-02-02 |website=www.numericana.com}}</ref>

== Symmetry ==
There are many symmetric forms, and the ones with highest symmetry have chiral [[icosahedral symmetry]]:

Four [[Catalan solid]]s, convex:

* [[Triakis octahedron]] - isosceles triangles
* [[Tetrakis hexahedron]] - isosceles triangles
* [[Deltoidal icositetrahedron]] - kites
* [[Pentagonal icositetrahedron]] - pentagons

27 [[List of uniform polyhedra|uniform star-polyhedral duals]]: (self-intersecting)

* [[Small rhombihexacron]], [[Great rhombihexacron]]
* [[Small hexacronic icositetrahedron]], [[Great hexacronic icositetrahedron]]
* [[Great deltoidal icositetrahedron]]
* [[Great triakis octahedron]]

Examples with lower symmetry include certain [[Dual polyhedron|dual polyhedra]] of [[Johnson solid]]s, such as the [[gyroelongated square bicupola]] and the [[elongated square gyrobicupola]].

== Common examples ==
Common examples include [[Prism (geometry)|prisms]] and [[Pyramid (geometry)|pyramids]], and include certain Johnson solids and [[Catalan solid]]s.

=== Icositrigonal pyramids ===
'''Icositrigonal pyramids''' are a type of cone with an [[icositrigon]] as a [[Base (geometry)|base]], with 24 [[Face (geometry)|faces]], 46 [[Edge (geometry)|edges,]] and 24 [[Vertex (geometry)|vertices]].<ref name=":02">{{Cite web |title=Icositrigonal pyramid - Wolfram{{!}}Alpha |url=https://www.wolframalpha.com/input/?i=Icositrigonal+pyramid |archive-url=http://web.archive.org/web/20241130162738/https://www.wolframalpha.com/input/?i=Icositrigonal+pyramid |archive-date=2024-11-30 |access-date=2025-02-02 |website=www.wolframalpha.com |language=en}}</ref> Regular icositrigonal pyramids have a regular icositrigon as a base, and its [[Schläfli symbol]] is {}∨{23}. The surface area <math>S</math> and volume <math>V</math> with side length <math>s</math> and height <math>h</math> can be calculated as follows:<ref name=":02" />

: <math>V=\frac{23 h s^2 \cot{\frac{\pi}{23}}}{12}\approx 13.9448 h s^2</math>

: <math>S=\frac{23s\left(\sqrt{4h^2+s^2\cot^{2}{\frac{\pi}{23}}}+s\cot{\frac{\pi}{23}}\right)}{4}\approx 5.75s\left(\sqrt{4h^2+52.9335s^2}+7.27554s\right)</math>

=== Icosidigonal prism ===
'''Icosidigonal prisms''' are a type of cylinder with an icosidigon as a base, with 24 faces, 66 edges, and 44 vertices.<ref>{{Cite web |title=Icosidigonal prism |url=https://www.wolframalpha.com/input/?i=Icosidigonal+prism |access-date=2025-02-02 |website=Wolfram Alpha}}</ref> Regular icosidigonal prisms have a regular icosidigon as a base, with each face a [[rectangle]]. Every vertex borders 2 squares and an icosidigon base. Its [[vertex configuration]] is <math>4{.}4{.}22</math>, its Schläfli symbol is {22}×{} or t{2,22}, its [[Coxeter–Dynkin diagram|Coxeter diagram]] is {{CDD|node_1|2x|2x|node|2|node_1}}, and its [[Conway polyhedron notation]] is P22. The surface area <math>S</math> and volume <math>V</math> with side length <math>s</math> and height <math>h</math> can be calculated as follows:

: <math>V=\frac{11 h s^2 \cot{\frac{\pi}{22}}}{2}\approx 38.2533 h s^2</math>

: <math>S=11s\left( 2h+s\cot{\frac{\pi}{22}}\right)\approx 11s\left(2h+6.95515s\right)</math>

=== Hendecagonal antiprism ===
[[File:Hendecagonal_antiprism.png|thumb|A hendecagonal antiprism]]
'''Hendecagonal antiprisms''' are [[antiprism]]s with a [[hendecagon]] as a base, with 24 faces, 44 edges, and 22 vertices. Regular hendecagonal antiprisms have a regular hendecagon as a base, with each face an equilateral triangle. Every vertex borders 2 triangles and a hendecagon base. Its vertex configuration is <math>11{.}3{.}3{.}3</math>.

=== Dodecagonal trapezohedron ===
'''Dodecagonal trapezohedra''' are the tenth member of the [[Trapezohedron|trapezohedra]] family, made of 24 [[Congruence (geometry)|congruent]] kites arranged radially. Every dodecagonal trapezohedron has 24 faces, 28 edges, and 26 vertices. There are two types of vertices, ones bordering 12 kits and ones bordering 3. Its dual polyhedron is the Hendecagonal antiprism.<ref>{{Cite web |title=12-trapezohedron |url=https://www.wolframalpha.com/input/?i=12-trapezohedron |access-date=2025-02-02 |website=Wolfram Alpha}}</ref> Its Schläfli symbol is { }⨁{12},  its Coxeter diagram is {{CDD||node_fh|2|node_fh|2x|4|node}} or {{CDD||node_fh|2|node_fh|12|node_fh}}, and its Conway polyhedron notation is dA12.

Dodecagonal trapezohedra are [[isohedral figure]]s.

=== Johnson solids ===
{{main|Johnson solid}}
There are two examples of Johnson solids which are icositetrahedra. They are listed as follows:
{| class="wikitable sortable"
!Name
!Image
!Designation
!Vertices
!Edges
!Faces
!Types of faces
!Symmetry group
!Net
|-
|[[Disphenocingulum]]
|[[File:Disphenocingulum.png|100x100px]]
|J<sub>90</sub>
|16
|38
|24
|20 equilateral triangle,
4 squares
|''D''<sub>2d</sub>
|[[File:Johnson_solid_90_net.png|100x100px]]
|-
|[[Triaugmented dodecahedron]]
|[[File:Triaugmented_dodecahedron.png|100x100px]]
|J<sub>61</sub>
|23
|45
|24
|15 equilateral triangles,
9 pentagons
|''C''<sub>3v</sub>
|[[File:Johnson_solid_61_net.png|100x100px]]
|}

=== Catalan Solids ===
{{main|Catalan solid}}
There are 5 types of  icositetrahedra with different [[Topology|topologies]].<ref>{{Citation |title=Sur la théorie des quantités positives et négatives |date=2009-07-20 |work=Cours d'analyse de l'École Royale Polytechnique |pages=403–437 |url=https://doi.org/10.1017/cbo9780511693328.016 |access-date=2025-02-02 |publisher=Cambridge University Press}}</ref> The pentagonal icositetetrahedron has two mirror images ([[Chirality (mathematics)|enantiomorphs]]), so geometrically there are 4 distinct Catalan icositetetrahedra.
{| class="wikitable sortable"
!Name
!Image
!Net
!Dual
!Faces
!Edges
!Vertices
![[Vertex configuration#Face configuration|Face Configuration]]
![[List of spherical symmetry groups#Polyhedral symmetry|Point Group]]
|-
|[[Triakis octahedron]]
|[[File:Triakisoctahedron.jpg|88x88px|Triakis octahedron]]
<small>([[:File:Triakisoctahedron.gif|animation]])</small>
|[[File:Triakisoctahedron_net.png|80x80px]]
|[[Truncated cube]]
| align="center" |24
| align="center" |36
| align="center" |14
| align="center" |Isosceles triangle
V3.8.8
| align="center" |O<sub>h</sub>
|-
|[[Tetrakis hexahedron]]
|[[File:Tetrakishexahedron.jpg|80x80px|Tetrakis hexahedron]]
<small>([[:File:Tetrakishexahedron.gif|animation]])</small>
|[[File:Tetrakishexahedron_net.png|80x80px]]
|[[Truncated octahedron]]
| align="center" |24
| align="center" |36
| align="center" |14
| align="center" |Isosceles triangle
V4.6.6
| align="center" |O<sub>h</sub>
|-
|[[Deltoidal icositetrahedron]]
|[[File:Deltoidalicositetrahedron.jpg|80x80px|Deltoidal icositetrahedron]]
<small>([[:File:Deltoidalicositetrahedron.gif|animation]])</small>
|[[File:Deltoidalicositetrahedron_net.png|80x80px]]
|[[Rhombicuboctahedron]]
| align="center" |24
| align="center" |48
| align="center" |26
| align="center" |Kite
V3.4.4.4
| align="center" |O<sub>h</sub>
|-
|[[Pentagonal icositetrahedron]]
|[[File:Pentagonalicositetrahedronccw.jpg|80x80px|Pentagonal icositetrahedron (Ccw)]]
<small>([[:File:Pentagonalicositetrahedronccw.gif|animation]])</small>
[[File:Pentagonalicositetrahedroncw.jpg|80x80px|Pentagonal icositetrahedron (Cw)]]
<small>([[:File:Pentagonalicositetrahedroncw.gif|animation]])</small>
|[[File:Pentagonalicositetrahedron_net.svg|80x80px]]
|[[Snub cube]]
| align="center" |24
| align="center" |60
| align="center" |38
| align="center" |irregular pentagon
V3.3.3.3.4
| align="center" |O
|}

=== Uniform star polyhedra ===
{{main|Uniform star polyhedron}}
Some uniform star polyhedra also have 24 faces:
{| class="wikitable sortable" style="text-align:center"
!Name
!Image
![[Wythoff symbol]]
![[Vertex figure]]
!Symmetry group
!Faces
!Edges
!Vertices
![[Euler characteristic]]
![[Density (polytope)|Density]]
!Faces by sides
|-
|[[Ditrigonal dodecadodecahedron]]
|[[File:Ditrigonal_dodecadodecahedron.png|60x60px]]
|3 {{pipe}} 5/3 5
|[[File:Ditrigonal_dodecadodecahedron_vertfig.png|50x50px]]
(5.5/3)<sup>3</sup>
|I<sub>h</sub>
|24
|60
|20
| -16
|4
|12{5}+12{5/2}
|-
|[[Dodecadodecahedron]]
|[[File:Dodecadodecahedron.png|60x60px]]
|5 5/2
|[[File:Dodecadodecahedron_vertfig.png|50x50px]]
5.5/2.5.5/2
|I<sub>h</sub>
|24
|60
|20
| -16
|4
|12{5}+12{5/2}
|-
|[[Truncated great dodecahedron]]
|[[File:Great_truncated_dodecahedron.png|60x60px]]
|2 5/2 {{pipe}} 5
|[[File:Truncated_great_dodecahedron_vertfig.png|50x50px]]
10.10.5/2
|I<sub>h</sub>
|24
|90
|60
| -6
|3
|12{5/2}+12{10}
|-
|[[Small stellated truncated dodecahedron]]
|[[File:Small_stellated_truncated_dodecahedron.png|60x60px]]
|2 5 {{pipe}} 5/3
|[[File:Small_stellated_truncated_dodecahedron_vertfig.png|50x50px]]
<sup>10</sup>/<sub>3</sub>.<sup>10</sup>/<sub>3</sub>.5
|I<sub>h</sub>
|24
|90
|60
| -6
|9
|12{5}+12{10/3}
|}

=== Types of icositetrahedra ===
{| class="wikitable sortable"
!Name
!Type
!Image
!Identifier
!Faces
!Edges
!Vertices
!Euler characteristic
!Types of faces
!Symmetry
!Net
|-
!Icosidigonal prism
|Prism
|
|t{2,22}
{22}x{}
{{CDD|node_1|2|node_1|2x|2x|node}}
|24
|66
|44
|2
|2 icosidigons,
22 squares
|D<sub>22h</sub>, [22,2], (*22 2 2), order 88
|
|-
!Icositrigonal pyramid
|Pyramid
|
|( )∨{23}
|24
|46
|24
|2
|1 icositrigon,
23 triangles
|''C''<sub>23v</sub>, [23], (*23 23)
|
|-
!Icosidigonal '''frustum'''
|[[Frustum]]
|
|
|24
|66
|44
|2
|2 icosidigons, 
22 trapezoids
|D<sub>22h</sub>, [22,2], (*22 2 2), order 88
|
|-
!Dodecagonal bipyramid
|[[Bipyramid]]
|
|{ } + {12}
{{CDD|node_f1|2|node_f1|12|node}}
|24
|36
|14
|2
|12 triangles
|D<sub>12h</sub>, [12,2], (*2 2 12), order 48
|
|-
!Dodecagonal trapezohedron
|Trapezohedron
|[[File:Dodecagonal_trapezohedron.png|130x130px]]
|{ }⨁{{mset|12}}<ref>{{cite book |last=Johnson |first=N.W. |title=Geometries and Transformations |year=2018 |isbn=978-1-107-10340-5 |page=235 |chapter=Chapter 11: Finite symmetry groups}}</ref>
|24
|48
|26
|2
|24 kites
|D<sub>12d</sub>, [2<sup>+</sup>,12], (2*12)
|
|-
!Hendecagonal antiprism
|Antiprism
|[[File:Hendecagonal_antiprism.png|100x100px]]
|s{2,22}
sr{2,11}
{{CDD|node_h|2x|node_h|2x|2x|node}}
{{CDD|node_h|2x|node_h|11|node_h}}
|24
|44
|22
|2
|2 hendecagons, 
22 triangles
|D<sub>11d</sub>, [2<sup>+</sup>,22], (2*11), order 44
|
|-
!Hendecagonal cupola
|[[Cupola (geometry)|Cupola]]
|
|
|24
|55
|33
|2
|11 equilateral triangles, 
11 squares, 
1 regular hendecagon,
1 regular icosidigon
|D<sub>11d</sub>, [2<sup>+</sup>,22], (2*11), order 44
|
|-
!Deltoidal icositetrahedron
|Johnson solid
|[[File:Deltoidalicositetrahedron.jpg|100x100px]]
|
|24
|48
|26
|2
|24 kites
|''D<sub>4d</sub>''
|[[File:Pseudo-strombic_icositetrahedron_flat.png|100x100px]]
|}

== See also ==

* [[Icositetragon]]

== References ==
<references />
*{{Mathworld |urlname=Icositetrahedron |title=Icositetrahedron}}
{{Polyhedra}}

[[Category:Polyhedra]]


{{Polyhedron-stub}}