Triakis octahedron
| Triakis octahedron | |
|---|---|
 (Click here for rotating model) | |
| Type | Catalan solid |
| Coxeter diagram |    
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| Conway notation | kO |
| Face type | V3.8.8 isosceles triangle |
| Faces | 24 |
| Edges | 36 |
| Vertices | 14 |
| Vertices by type | 8{3}+6{8} |
| Symmetry group | Oh,B3,[4,3], (*432) |
| Rotation group | O, [4,3]+,(432) |
| Dihedral angle | 147°21′00″ arccos(−3 + 8√2/17) |
| Properties | convex,face-transitive |
 Truncated cube (dual polyhedron) |
 Net |
Ingeometry,atriakis octahedron(ortrigonal trisoctahedron[1]orkisoctahedron[2]) is anArchimedean dualsolid, or aCatalan solid.Its dual is thetruncated cube.
It can be seen as anoctahedronwithtriangular pyramidsadded to each face; that is, it is theKleetopeof the octahedron. It is also sometimes called atrisoctahedron,or, more fully,trigonal trisoctahedron.Both names reflect that it has three triangular faces for every face of an octahedron. Thetetragonal trisoctahedronis another name for thedeltoidal icositetrahedron,a different polyhedron with three quadrilateral faces for every face of an octahedron.
This convex polyhedron is topologically similar to the concavestellated octahedron.They have the same face connectivity, but the vertices are at different relative distances from the center.
If its shorter edges have length of 1, its surface area and volume are:
Cartesian coordinates
[edit]Letα=√2− 1,then the 14 points(±α,±α,±α)and(±1, 0, 0),(0, ±1, 0)and(0, 0, ±1)are the vertices of a triakis octahedron centered at the origin.
The length of the long edges equals√2,and that of the short edges2√2− 2.
The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equalsarccos(1/4−√2/2)≈117.20057038016° and the acute ones equalarccos(1/2+√2/4)≈31.39971480992°.
Orthogonal projections
[edit]Thetriakis octahedronhas three symmetry positions, two located on vertices, and one mid-edge:
| Projective symmetry |
[2] | [4] | [6] |
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| Triakis octahedron |
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| Truncated cube |
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Cultural references
[edit]- A triakis octahedron is a vital element in the plot of cult authorHugh Cook's novelThe Wishstone and the Wonderworkers.
Related polyhedra
[edit]The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
| Uniform octahedral polyhedra | ||||||||||
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| Symmetry:[4,3],(*432) | [4,3]+ (432) |
[1+,4,3] = [3,3] (*332) |
[3+,4] (3*2) | |||||||
| {4,3} | t{4,3} | r{4,3} r{31,1} |
t{3,4} t{31,1} |
{3,4} {31,1} |
rr{4,3} s2{3,4} |
tr{4,3} | sr{4,3} | h{4,3} {3,3} |
h2{4,3} t{3,3} |
s{3,4} s{31,1} |
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| V43 | V3.82 | V(3.4)2 | V4.62 | V34 | V3.43 | V4.6.8 | V34.4 | V33 | V3.62 | V35 |
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The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. Theseface-transitivefigures have (*n32) reflectionalsymmetry.
| *n32 symmetry mutation of truncated tilings: t{n,3} | |||||||||||
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| Symmetry *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | ||||||
| *232 [2,3] |
*332 [3,3] |
*432 [4,3] |
*532 [5,3] |
*632 [6,3] |
*732 [7,3] |
*832 [8,3]... |
*∞32 [∞,3] |
[12i,3] | [9i,3] | [6i,3] | |
| Truncated figures |
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| Symbol | t{2,3} | t{3,3} | t{4,3} | t{5,3} | t{6,3} | t{7,3} | t{8,3} | t{∞,3} | t{12i,3} | t{9i,3} | t{6i,3} |
| Triakis figures |
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| Config. | V3.4.4 | V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12 | V3.14.14 | V3.16.16 | V3.∞.∞ | |||
The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. Theseface-transitivefigures have (*n42) reflectionalsymmetry.
| *n42 symmetry mutation of truncated tilings: n.8.8 | |||||||||||
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| Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | |||||||
| *242 [2,4] |
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] | ||||
| Truncated figures |
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| Config. | 2.8.8 | 3.8.8 | 4.8.8 | 5.8.8 | 6.8.8 | 7.8.8 | 8.8.8 | ∞.8.8 | |||
| n-kis figures |
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| Config. | V2.8.8 | V3.8.8 | V4.8.8 | V5.8.8 | V6.8.8 | V7.8.8 | V8.8.8 | V∞.8.8 | |||
References
[edit]- ^"Clipart tagged: 'forms'".etc.usf.edu.
- ^Conway, Symmetries of things, p. 284
- Williams, Robert(1979).The Geometrical Foundation of Natural Structure: A Source Book of Design.Dover Publications, Inc.ISBN0-486-23729-X.(Section 3-9)
- Wenninger, Magnus(1983),Dual Models,Cambridge University Press,doi:10.1017/CBO9780511569371,ISBN978-0-521-54325-5,MR0730208(The thirteen semiregular convex polyhedra and their duals, Page 17, Triakisoctahedron)
- The Symmetries of Things2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,ISBN978-1-56881-220-5[1](Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis octahedron)
External links
[edit]- Weisstein, Eric W.,"Triakis octahedron"("Catalan solid") atMathWorld.
- Triakis Octahedron– Interactive Polyhedron Model
- Virtual Reality Polyhedrawww.georgehart.com: The Encyclopedia of Polyhedra
- VRMLmodel
- Conway Notation for PolyhedraTry: "dtC"
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