Editing Dodecahedron (section)

===Tetartoid===
{| class="wikitable floatright" style="width:260px;"
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! style="background:#e7dcc3;" colspan="2"|Tetartoid<br>Tetragonal pentagonal dodecahedron
|-
| style="text-align:center;" colspan="2"|[[File:Tetartoid perspective.png|250px]]<br><small>(See [[c:File:Tetartoid perspective.gif|here]] for a rotating model.)</small>
|-
| style="background:#e7dcc3;"|Face polygon||[[pentagon|irregular pentagon]]
|-
| style="background:#e7dcc3;"|[[Conway polyhedron notation|Conway notation]]||gT
|-
| style="background:#e7dcc3;"|[[Face (geometry)|Faces]]||12
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| style="background:#e7dcc3;"|[[Edge (geometry)|Edges]]||30 (6+12+12)
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| style="background:#e7dcc3;"|[[Vertex (geometry)|Vertices]]||20 (4+4+12)
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| style="background:#e7dcc3;"|[[List of spherical symmetry groups#Polyhedral sym|Symmetry group]]||[[tetrahedral symmetry|T]], [3,3]<sup>+</sup>, (332), order 12
<!--|-
|bgcolor=#e7dcc3|[[Dual polyhedron]]||[[Pseudoicosahedron]]-->
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| style="background:#e7dcc3;"|Properties||[[convex set|convex]], [[face transitive]]
<!--|- align=center
|colspan=2|[[Net (polyhedron)|Net]]<BR>[[File:Pyritohedron flat.png|200px]]-->
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A '''tetartoid''' (also '''tetragonal pentagonal dodecahedron''', '''pentagon-tritetrahedron''', and '''tetrahedric pentagon dodecahedron''') is a dodecahedron with chiral [[tetrahedral symmetry]] (T). Like the [[regular dodecahedron]], it has twelve identical [[pentagon]]al faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes.

Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry.<ref>Dutch, Steve. [https://www.uwgb.edu/dutchs/symmetry/xlforms.htm The 48 Special Crystal Forms] {{Webarchive|url=https://web.archive.org/web/20130918103121/https://www.uwgb.edu/dutchs/symmetry/xlforms.htm |date=2013-09-18 }}. Natural and Applied Sciences, [[University of Wisconsin-Green Bay]], U.S.</ref> The mineral [[cobaltite]] can have this symmetry form.<ref>[http://www.galleries.com/minerals/property/crystal.htm#dodecahe Crystal Habit]. Galleries.com. Retrieved on 2016-12-02.</ref>

Abstractions sharing the solid's [[topology]] and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In [[Conway polyhedron notation]] this is a gyro tetrahedron.)

{|
|
{{multiple image |align=left |total_width=440
| image1 = Tetartoid from red.png
| image2 = Tetartoid from green.png
| image3 = Tetartoid from yellow.png
| footer = Orthographic projections from 2- and 3-fold axes
}}
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{{multiple image |align=left |total_width=300
| image1 = Tetartoid cube.png
| image2 = Tetartoid tetrahedron.png
| footer = Cubic and tetrahedral form
}}
|
[[File:Cobaltite-d05-67a.jpg|143px|thumb|[[Cobaltite]]]]
|}

{| class="wikitable collapsible collapsed"
! Relationship to the dyakis dodecahedron
|-
|style="width: 760px;"|
A tetartoid can be created by enlarging 12 of the 24 faces of a [[dyakis dodecahedron]].
(The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the [[disdyakis dodecahedron]].)
<!--start inner table-->
{|
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{{multiple image |align=left |total_width=550
| image1 = Tetartoid dark vertical (with traces of dyakis 12).png
| image2 = Disdyakis 12 untruncated to dyakis 12 vertical.png
| image3 = Tetartoid light vertical (with traces of dyakis 12).png
| footer = [[Chirality|Chiral]] tetartoids based on the dyakis dodecahedron in the middle
}}
|
[[File:Crystal model of tetartoid around dyakis dodecahedron (mirrored).jpg|thumb|right|155px|Crystal model]]
|}<!--end inner table-->
The [[crystal model]] on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges.
|}

====Cartesian coordinates====
The following points are vertices of a tetartoid pentagon under [[tetrahedral symmetry]]:
:(''a'', ''b'', ''c''); (−''a'', −''b'', ''c''); (−{{sfrac|''n''|''d''<sub>1</sub>}}, −{{sfrac|''n''|''d''<sub>1</sub>}}, {{sfrac|''n''|''d''<sub>1</sub>}}); (−''c'', −''a'', ''b''); (−{{sfrac|''n''|''d''<sub>2</sub>}}, {{sfrac|''n''|''d''<sub>2</sub>}}, {{sfrac|''n''|''d''<sub>2</sub>}}),
under the following conditions:<ref>[http://demonstrations.wolfram.com/TheTetartoid/ The Tetartoid]. Demonstrations.wolfram.com. Retrieved on 2016-12-02.</ref>
:{{nowrap|1=0 ≤ ''a'' ≤ ''b'' ≤ ''c''}},
:''n'' = ''a''<sup>2</sup>''c'' − ''bc''<sup>2</sup>,
:''d''<sub>1</sub> = ''a''<sup>2</sup> − ''ab'' + ''b''<sup>2</sup> + ''ac'' − 2''bc'',
:''d''<sub>2</sub> = ''a''<sup>2</sup> + ''ab'' + ''b''<sup>2</sup> − ''ac'' − 2''bc'',
:{{nowrap|1=''nd''<sub>1</sub>''d''<sub>2</sub> ≠ 0}}.

====Geometric freedom====

The [[regular dodecahedron]] is a tetartoid with more than the required symmetry. The [[triakis tetrahedron]] is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.)

{| class="wikitable collapsible collapsed"
!colspan="8"| Tetartoid variations from [[regular dodecahedron]] to [[triakis tetrahedron]]
|- style="background-color: white;"
|[[File:Dodecahedron.png|140px]]
|[[File:Tetartoid-010.png|150px]]
|[[File:Tetartoid-020.png|150px]]
|[[File:Tetartoid-040.png|150px]]
|[[File:Tetartoid-060.png|150px]]
|[[File:Tetartoid-080.png|150px]]
|[[File:Tetartoid-095.png|150px]]
|[[File:Triakistetrahedron.jpg|100px]]
|}
{{Clear}}