View source for Deltoidal icositetrahedron


{{Short description|Catalan solid with 24 kite faces}}

{| class=wikitable align="right" width="250"
!bgcolor=#e7dcc3 colspan=2|Deltoidal icositetrahedron
|-
|align=center colspan=2|[[Image:deltoidalicositetrahedron.jpg|290px|Deltoidal icositetrahedron]]<br>([[:Image:deltoidalicositetrahedron.gif|rotating]] and [[:File:Deltoidal icositetrahedron.stl|3D model]]) 
|-
|bgcolor=#e7dcc3|Type||[[Catalan solid|Catalan]]
|-
|bgcolor=#e7dcc3|[[Conway polyhedron notation|Conway notation]]||oC or deC
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_f1|4|node|3|node_f1}}
|-
|bgcolor=#e7dcc3|Face polygon||[[File:DU10 facets.png|60px]]<BR>[[Kite (geometry)|Kite]] with 3 equal acute angles & 1 obtuse angle
|-
|bgcolor=#e7dcc3|Faces||24, [[Congruence (geometry)|congruent]]
|-
|bgcolor=#e7dcc3|Edges||24 short + 24 long = 48
|-
|bgcolor=#e7dcc3|Vertices||8 (connecting 3 short edges)<br>+ 6 (connecting 4 long edges)<br>+ 12 (connecting 4 alternate short & long edges)<br>= 26
|-
|bgcolor=#e7dcc3|[[Face configuration]]||V3.4.4.4
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||[[Octahedral symmetry|O<sub>h</sub>]], BC<sub>3</sub>, [4,3], *432
|-
|bgcolor=#e7dcc3|[[Point groups in three dimensions#Rotation groups|Rotation group]]||O, [4,3]<sup>+</sup>, (432)
|-
|bgcolor=#e7dcc3|[[Dihedral angle]]||same value for short & long edges:<br><math>\arccos \left( - \frac{7 + 4 \sqrt{2}}{17} \right)</math><br><math>\approx 138^{\circ} 07 ' 05 ''</math>
|-
|bgcolor=#e7dcc3|[[Dual polyhedron]]||[[Rhombicuboctahedron]]
|-
|bgcolor=#e7dcc3|Properties||convex, [[face-transitive]]
|-
|align=center colspan=2|[[Image:Deltoidalicositetrahedron net.png|150px|Deltoidal icositetrahedron]]<br>[[Net (polyhedron)|Net]]
|}

{{multiple image
| align             = right
| total_width       = 300
| image1            = Eeuwig water, Weiwerd, crop.jpg
| image2            = D24_deltoidal_icositetrahedron.JPG
| footer            = <abbr title="Deltoidal icositetrahedron">D.i.</abbr> as artwork and [[Dice|die]]
}}

{{multiple image
| align             = right
| total_width       = 300
| image1            = Perspectiva Corporum Regularium 36b.jpg
| image2            = Perspectiva Corporum Regularium 34b.jpg
| footer            = <abbr title="Deltoidal icositetrahedron">D.i.</abbr> projected onto cube and octahedron in ''[[Perspectiva Corporum Regularium]]''
}}

{{multiple image
| align             = right
| total_width       = 300
| image1            = Modell, Kristallform Disdodekaeders (Dyakisdodekaeder) -Krantz 374-.jpg
| image2            = Deltoidal icositetrahedron octahedral gyro.png
| footer            = Dyakis dodecahedron [[crystal model]] and projection onto octahedron
}}

In [[geometry]], the '''deltoidal icositetrahedron''' (or '''trapezoidal icositetrahedron''', '''tetragonal icosikaitetrahedron''',<ref>Conway, ''Symmetries of Things'', p. 284–286.</ref> '''tetragonal trisoctahedron''',<ref>{{Cite web|url=https://etc.usf.edu/clipart/keyword/forms|title = Keyword: "forms" &#124; ClipArt ETC}}</ref> '''strombic icositetrahedron''') is a [[Catalan solid]]. Its 24 faces are [[Congruence (geometry)|congruent]] [[Kite (geometry)|kites]].<ref>{{cite web |title=Kite |url=https://www.mathsisfun.com/geometry/kite.html |access-date=6 October 2019}}</ref> The deltoidal icositetrahedron, whose [[Dual polyhedron|dual]] is the (uniform) [[rhombicuboctahedron]], is tightly related to the pseudo-deltoidal icositetrahedron, whose dual is the [[pseudorhombicuboctahedron]]; but the actual and pseudo-d.i. are not to be confused with each other.

== Cartesian coordinates ==
[[File:Polyhedron small rhombi 6-8 dual max.png|290px]]

In the image above, the long body diagonals are those between opposite red vertices and between opposite blue vertices, and the short body diagonals are those between opposite yellow vertices.<br>[[Cartesian coordinates]] for the vertices of the deltoidal icositetrahedron centered at the origin and with long body diagonal length 2 are:
*red vertices (lying in <math>4</math>-fold symmetry axes):
:<math>\left( \pm 1 , 0 , 0 \right) , \left( 0 , \pm 1 , 0 \right) , \left( 0 , 0 , \pm 1 \right) ;</math>
*blue vertices (lying in <math>2</math>-fold symmetry axes):
:<math>\left( 0 , \pm \frac{\sqrt{2}}{2} , \pm \frac{\sqrt{2}}{2} \right) , \left( \pm \frac{\sqrt{2}}{2} , 0 , \pm \frac{\sqrt{2}}{2} \right) , \left( \pm \frac{\sqrt{2}}{2} , \pm \frac{\sqrt{2}}{2} , 0 \right) ;</math>
*yellow vertices (lying in <math>3</math>-fold symmetry axes):
:<math>\left( \pm \frac{2 \sqrt{2} + 1}{7} , \pm \frac{2 \sqrt{2} + 1}{7} , \pm \frac{2 \sqrt{2} + 1}{7} \right) .</math>
Where <math> \sqrt{2} + 1 </math> is the '''silver ratio''', δs. 

For example, the point with coordinates <math>\left( \frac{2 \sqrt{2} + 1}{7} , \frac{2 \sqrt{2} + 1}{7} , \frac{2 \sqrt{2} + 1}{7} \right)</math> is the intersection of the plane with equation <math>\left( \sqrt{2} - 1 \right) x + \left( \sqrt{2} - 1 \right) y + 1 \left( z - 1 \right) = 0</math> and of the line with system of equations <math>x = y = z\,.</math>

A deltoidal icositetrahedron has three regular-octagon equators, lying in three orthogonal planes.

== Dimensions and angles ==
=== Dimensions ===
The deltoidal icositetrahedron with long body diagonal length ''D'' = 2 has:
*short body diagonal length:
::<math>d = \frac{2 \sqrt{3} \left( 2 \sqrt{2} + 1 \right) }{7} \approx 1.894\,580 ;</math>
*long edge length:<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Deltoidal Icositetrahedron |url=https://mathworld.wolfram.com/DeltoidalIcositetrahedron.html |access-date=2022-10-06 |website=mathworld.wolfram.com |language=en}}<br />In this MathWorld entry, the small rhombicuboctahedron has edge length <math>a = 1 ;</math> so this s.r.c.o.h. has circumradius <math>R = \frac{ \sqrt{5 + 2 \sqrt{2}} }{2}</math> and midradius <math>\rho = \sqrt{1 + \frac{ \sqrt{2} }{2}} ;</math> so this s.r.c.o.h.'s dual with respect to their common midsphere is the deltoidal icositetrahedron with inradius <math>r _{ _{ a = 1 } } = \frac{ \rho ^{2} }{R} = \sqrt{ \frac{ 2 \left( 7 + 4 \sqrt{2} \right) }{17} } = \sqrt{2}</math>×<math>\sqrt{ \frac{7 + 4 \sqrt{2}}{17} } = \sqrt{2}</math>×<math>r _{ _{ D = 2 } } .</math></ref>
::<math>S = \sqrt{2 - \sqrt{2}} \approx 0.765\,367 ;</math>
*short edge length:<ref name=":0" />
::<math>s = \frac{ \sqrt{20 - 2 \sqrt{2}} }{7} \approx 0.591\,980 ;</math>
*inradius:<ref name=":0" />
::<math>r = \sqrt{ \frac{7 + 4 \sqrt{2}}{17} } \approx 0.862\,856 .</math>

<math>r</math> is the distance from the center to any face plane; it may be calculated by [[Plane (geometry)#Operations##Distance from a point to a plane|normalizing]] the equation of plane [[Deltoidal icositetrahedron#Cartesian coordinates|above]], replacing (''x'', ''y'', ''z'') with (0, 0, 0), and taking the [[absolute value]] of the result.

A deltoidal icositetrahedron has its long and short edges in the ratio:
:<math>\frac{ S }{ s } = 2 - \frac{1}{ \sqrt{2} } \approx 1.292\,893 .</math>

The deltoidal icositetrahedron with short edge length <math>s</math> has:
*area:<ref name=":0" />
::<math>A = 6 \sqrt{29 - 2 \sqrt{2}}\,s ^{2} ;</math>
*volume:<ref name=":0" />
::<math>V = \sqrt{122 + 71 \sqrt{2}}\,s ^{3} .</math>

=== Angles ===
For a deltoidal icositetrahedron, each kite face has:
*three equal acute angles, with value:
::<math>\arccos \left( \frac{1}{2} - \frac{ \sqrt{2} }{4} \right) \approx 81.578\,942^{\circ} ;</math>
*one obtuse angle (between the short edges), with value:
::<math>\arccos \left( - \frac{1}{4} - \frac{ \sqrt{2} }{8} \right) \approx 115.263\,174^{\circ} .</math>

=== Side Lengths ===
In a deltoidal icositetrahedron, each face is a kite-shaped quadrilateral. The side lengths of these kites can be expressed in the ratio 0.7731900694928638:1
Specifically, the side adjacent to the obtuse angle has a length of approximately 0.707106785, while the side adjacent to the acute angle has a length of approximately 0.914213565.

== Occurrences in nature and culture ==
The deltoidal icositetrahedron is a [[crystal habit]] often formed by the mineral [[analcime]] and occasionally [[garnet]]. The shape is often called a trapezohedron in mineral contexts, although in [[solid geometry]] the name [[trapezohedron]] has another meaning.

In Guardians of The Galaxy Vol. 3, the device containing the files about the experiments carried on Rocket Raccoon has the shape of a deltoidal icositetrahedron.

== Orthogonal projections ==
The ''deltoidal icositetrahedron'' has three symmetry positions, all centered on vertices:
{|class=wikitable
|+ Orthogonal projections
|- align=center
!Projective<BR>symmetry
|[2]
|[4]
|[6]
|-
!Image
|[[File:Dual cube t02 f4b.png|120px]]
|[[File:Dual cube t02 B2.png|120px]]
|[[File:Dual cube t02.png|120px]]
|-
!Dual<BR>image
|[[File:Cube t02 f4b.png|120px]]
|[[File:3-cube t02 B2.svg|120px]]
|[[File:3-cube t02.svg|120px]]
|}

== Related polyhedra ==
The deltoidal icositetrahedron's projection onto a [[cube]] divides its squares into quadrants. The projection onto a regular [[octahedron]] divides its equilateral triangles into kite faces. In [[Conway polyhedron notation]] this represents an ''ortho'' operation to a cube or octahedron. 

The deltoidal icositetrahedron <small>(dual of the [[Rhombicuboctahedron|small rhombicuboctahedron]])</small> is tightly related to the [[disdyakis dodecahedron]] <small>(dual of the [[Truncated cuboctahedron|great rhombicuboctahedron]])</small>. The main difference is that the latter also has edges between the vertices on 3- and 4-fold symmetry axes <small>(between yellow and red vertices in the images below)</small>.

{| class="wikitable" style="text-align: center;"
| [[File:Disdyakis 12 in deltoidal 24.png|140px|]]
| [[File:Disdyakis 12.png|140px|]]
| [[File:Disdyakis 12 untruncated to dyakis 12 horizontal with traces.png|140px|]]
| [[File:Tetartoid dark horizontal (with traces of dyakis 12).png|140px|]]
|-
| '''Deltoidal<br>icositetrahedron'''
| [[Disdyakis dodecahedron|Disdyakis<br>dodecahedron]]
| [[Dyakis dodecahedron|Dyakis<br>dodecahedron]]
| [[Tetartoid]]
|}

=== Dyakis dodecahedron ===

A variant with [[pyritohedral symmetry]] is called a '''[[dyakis dodecahedron]]'''<ref>[http://loki3.com/poly/isohedra.html#24k Isohedron 24k]</ref><ref>[http://metafysica.nl/isometric.html The Isometric Crystal System ]</ref> or '''diploid'''.<ref>[https://web.archive.org/web/20130918103121/https://www.uwgb.edu/dutchs/symmetry/xlforms.htm The 48 Special Crystal Forms]</ref> It is common in [[crystallography]].<br>A dyakis dodecahedron can be created by enlarging 24 of the 48 faces of a disdyakis dodecahedron. A [[tetartoid]] can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron.
[[File:Perfect_diploid.stl|thumb|3D model of a dyakis dodecahedron]]<ref>Both is indicated in the two crystal models in the top right corner of [[:File:Modèles cristallographiques-Musée de minéralogie de Strasbourg.jpg|this photo]]. A visual demonstration can be seen [[c:Category:Disdyakis polyhedra untruncated (light and dark brown)#Transitions|here]] and [[c:Category:Tetartoid (light and dark brown)#Transitions|here]].</ref>

=== Stellation ===
The [[great triakis octahedron]] is a stellation of the deltoidal icositetrahedron.

== Related polyhedra and tilings ==
The deltoidal icositetrahedron is a member of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

When projected onto a sphere (see right), it can be seen that the edges make up [[Compound of cube and octahedron|the edges of a cube and regular octahedron arranged in their dual positions]]. It can also be seen that the 3- and 4-fold corners can be made to have the same distance to the center. In that case the resulting icositetrahedron will no longer have a rhombicuboctahedron for a dual, since the centers of the square and triangle faces of a rhombicuboctahedron are at different distances from its center.

{{Octahedral truncations}}

This polyhedron is a term of a sequence of topologically related deltoidal polyhedra with face configuration V3.4.''n''.4; this sequence continues with tilings of the [[Euclidean space|Euclidean]] and [[Hyperbolic space|hyperbolic]] planes. These [[face-transitive]] figures have (*''n''32) reflectional [[Orbifold notation|symmetry]].

{{Dual expanded table}}

== See also ==
* [[Deltoidal hexecontahedron]]
* [[Tetrakis hexahedron]], another 24-face Catalan solid which looks a bit like an overinflated cube.
* "[[The Haunter of the Dark]]", a story by H.P. Lovecraft, whose plot involves this figure.

== References ==
{{reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{Citation |last=Wenninger |first=Magnus |author-link=Magnus Wenninger |title=Dual Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-54325-5 |mr=730208 |year=1983 |doi=10.1017/CBO9780511569371}} (The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron)
*''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{ISBN|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)

== External links ==
*{{Mathworld2 |urlname=DeltoidalIcositetrahedron |title=Deltoidal icositetrahedron |urlname2=CatalanSolid |title2=Catalan solid}}
*[https://web.archive.org/web/20080613172751/http://polyhedra.org/poly/show/36/trapezoidal_icositetrahedron Deltoidal (Trapezoidal) Icositetrahedron] – Interactive Polyhedron model

{{Catalan solids}}
{{Polyhedron navigator}}

[[Category:Catalan solids]]