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Triakis truncated tetrahedral honeycomb image
 
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{{Short description|sixteen-sided polyhedron}}
{{Distinguish|truncated triakis tetrahedron}}
{{Distinguish|truncated triakis tetrahedron}}
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{{short description|Space-filling polyhedron with 16 faces}}
{| class=wikitable align=right width="250"
{{Infobox polyhedron
!bgcolor=#e7dcc3 colspan=2|Triakis truncated tetrahedron
| image = Triakis truncated tetrahedron.png
|-
| type = [[Plesiohedron]]
|align=center colspan=2|[[Image:Triakis truncated tetrahedron.png|240px|Triakis truncated tetrahedron]] [[:File:Triakis truncated tetrahedron.gif|(Click here for rotating model)]]
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| faces = 4 [[hexagon]]s<br>12 [[isosceles triangle]]s
|-
| edges = 30
|bgcolor=#e7dcc3|Type||[[Plesiohedron]]
| vertices = 16
|-
| vertex_config =
|bgcolor=#e7dcc3|[[Conway polyhedron notation|Conway notation]]||k3tT
| schläfli =
|-
| wythoff =
⚫
|bgcolor=#e7dcc3|Faces||4 [[hexagon]]s<br>12 [[isosceles triangle]]s
| conway = {{math|k3tT}}
|-
| coxeter =
|bgcolor=#e7dcc3|Edges||30
| symmetry =
|-
| rotation_group =
|bgcolor=#e7dcc3|Vertices||16
⚫
| dual = 16{{!}}Order-3 [[truncated triakis tetrahedron]]
|-
| properties = [[convex polytope|convex]]
⚫
|bgcolor=#e7dcc3|Dual||16|[[Order-3 truncated triakis tetrahedron]]
| vertex_figure =
|-
| net =
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], space-filling
|}
}}
In [[geometry]], the '''triakis truncated tetrahedron''' is a [[Convex polyhedron|convex]] [[polyhedron]] made from 4 hexagons and 12 isosceles triangles. It can be used to [[Tessellation|tessellate]] three-dimensional space, making the [[triakis truncated tetrahedral honeycomb]].<ref name=conway2008>{{cite book|last1=Conway|first1=John H.|last2=Burgiel|first2=Heidi|last3=Goodman-Strauss|first3=Chaim|title=The Symmetries of Things|page=332|year=2008|isbn=978-1568812205}}</ref><ref>{{cite journal|last1=Grünbaum|first1=B|last2=Shephard|first2=G. C.|title=Tilings with Congruent Tiles|journal=Bull. Amer. Math. Soc.|volume=3|issue=3|pages=951–973|year=1980|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183547682|doi=10.1090/s0273-0979-1980-14827-2|doi-access=free}}</ref>
In [[geometry]], the '''triakis truncated tetrahedron''' is a [[Convex polyhedron|convex]] [[polyhedron]] made from 4 hexagons and 12 [[isosceles triangle]]s. It can be used to [[Tessellation|tessellate]] three-dimensional space, making the [[triakis truncated tetrahedral honeycomb]].<ref name=conway2008>{{cite book|last1=Conway|first1=John H.|last2=Burgiel|first2=Heidi|last3=Goodman-Strauss|first3=Chaim|title=The Symmetries of Things|page=332|year=2008|isbn=978-1568812205}}</ref><ref>{{cite journal|last1=Grünbaum|first1=B|last2=Shephard|first2=G. C.|title=Tilings with Congruent Tiles|journal=Bull. Amer. Math. Soc.|volume=3|issue=3|pages=951–973|year=1980|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183547682|doi=10.1090/s0273-0979-1980-14827-2|doi-access=free}}</ref>


The triakis truncated tetrahedron is the shape of the [[Voronoi cell]] of the [[carbon]] atoms in [[diamond]], which lie on the [[diamond cubic]] crystal structure.<ref>{{cite journal|first1=L.|last1=Föppl|year=1914|title=Der Fundamentalbereich des Diamantgitters|journal=Phys. Z.|volume=15|pages=191–193}}</ref><ref name=conway2003>{{cite web|last=Conway|first=John|title=Voronoi Polyhedron|url=https://groups.google.com/forum/?fromgroups=#!msg/geometry.puzzles/pkL3avbWPoc/ABSaqdQaqu4J|work=geometry.puzzles|accessdate=20 September 2012}}</ref> As the Voronoi cell of a symmetric space pattern, it is a [[plesiohedron]].<ref>{{citation
The triakis truncated tetrahedron is the shape of the [[Voronoi cell]] of the [[carbon]] atoms in [[diamond]], which lie on the [[diamond cubic]] crystal structure.<ref>{{cite journal|first1=L.|last1=Föppl|year=1914|title=Der Fundamentalbereich des Diamantgitters|journal=Phys. Z.|volume=15|pages=191–193}}</ref><ref name=conway2003>{{cite web|last=Conway|first=John|title=Voronoi Polyhedron|url=https://groups.google.com/forum/?fromgroups=#!msg/geometry.puzzles/pkL3avbWPoc/ABSaqdQaqu4J|work=geometry.puzzles|accessdate=20 September 2012}}</ref> As the Voronoi cell of a symmetric space pattern, it is a [[plesiohedron]].<ref>{{citation
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==Construction==
==Construction==
[[File:Triakis truncated tetrahedral honeycomb.jpg|thumb|Triakis truncated tetrahedral honeycomb]]

For space-filling, the triakis truncated tetrahedron can be constructed as follows:
For space-filling, the triakis truncated tetrahedron can be constructed as follows:
# Truncate a regular [[tetrahedron]] such that the big faces are regular hexagons.
# Truncate a regular [[tetrahedron]] such that the big faces are regular hexagons.

Latest revision as of 06:31, 13 August 2022

Not to be confused with truncated triakis tetrahedron.
Triakis truncated tetrahedron
TypePlesiohedron
Faces4 hexagons
12 isosceles triangles
Edges30
Vertices16
Conway notationk3tT
Dual polyhedron16|Order-3 truncated triakis tetrahedron
Propertiesconvex

In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb.[1][2]

The triakis truncated tetrahedron is the shape of the Voronoi cell of the carbon atoms in diamond, which lie on the diamond cubic crystal structure.[3][4] As the Voronoi cell of a symmetric space pattern, it is a plesiohedron.[5]

Construction

[edit]
Triakis truncated tetrahedral honeycomb

For space-filling, the triakis truncated tetrahedron can be constructed as follows:

  1. Truncate a regular tetrahedron such that the big faces are regular hexagons.
  2. Add an extra vertex at the center of each of the four smaller tetrahedra that were removed.

See also

[edit]

References

[edit]
  1. ^ Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. p. 332. ISBN 978-1568812205.
  2. ^ Grünbaum, B; Shephard, G. C. (1980). "Tilings with Congruent Tiles". Bull. Amer. Math. Soc. 3 (3): 951–973. doi:10.1090/s0273-0979-1980-14827-2.
  3. ^ Föppl, L. (1914). "Der Fundamentalbereich des Diamantgitters". Phys. Z. 15: 191–193.
  4. ^ Conway, John. "Voronoi Polyhedron". geometry.puzzles. Retrieved 20 September 2012.
  5. ^ Grünbaum, Branko; Shephard, G. C. (1980), "Tilings with congruent tiles", Bulletin of the American Mathematical Society, New Series, 3 (3): 951–973, doi:10.1090/S0273-0979-1980-14827-2, MR 0585178.


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