Augmented truncated tetrahedron
Augmented truncated tetrahedron | |
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Type | Johnson J64 – J65 – J66 |
Faces | 8 triangles 3 squares 3 hexagons |
Edges | 27 |
Vertices | 15 |
Vertex configuration | 2x3(3.62) 3(3.4.3.4) 6(3.4.3.6) |
Symmetry group | C3v |
Properties | convex |
Net | |
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![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/J65_augmented_truncated_tetrahedron.stl/220px-J65_augmented_truncated_tetrahedron.stl.png)
In geometry, the augmented truncated tetrahedron is a polyhedron constructed by attaching a triangular cupola onto an truncated tetrahedron. It is an example of a Johnson solid.
Construction
The augmented truncated tetrahedron is constructed from a truncated tetrahedron by attaching a triangular cupola.[1] This cupola covers one of the truncated tetrahedron's four hexagonal faces, so that the resulting polyhedron's faces are eight equilateral triangles, three squares, and three regular hexagons.[2] Since it has the property of convexity and has regular polygonal faces, the augmented truncated tetrahedron is a Johnson solid, denoted as the sixty-fifth Johnson solid .[3]
Properties
The surface area of an augmented truncated tetrahedron is:[2] the sum of the areas of its faces. Its volume can be calculated by slicing it off into both truncated tetrahedron and triangular cupola, and adding their volume:[2]
It has the same three-dimensional symmetry group as the triangular cupola, the pyramidal symmetry . Its dihedral angles can be obtained by adding the angle of a triangular cupola and an augmented truncated tetrahedron in the following:[4]
- its dihedral angle between triangle and hexagon is as in the truncated tetrahedron: 109.47°;
- its dihedral angle between adjacent hexagons is as in the truncated tetrahedron: 70.53°;
- its dihedral angle between triangle and square is as in the triangular cupola's angle: 125.3°
- its dihedral angle between triangle and square, on the edge where the triangular cupola and truncated tetrahedron are attached, is the sum of both triangular cupola's square-hexagon angle and the truncated tetrahedron's triangle-hexagon angle: approximately 164.17°; and
- its dihedral angle between triangle and hexagon, on the edge where triangular cupola and truncated tetrahedron are attached, is the sum of the dihedral angle of a triangular cupola and truncated tetrahedron between that: approximately 141.3°;
References
- ^ Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
- ^ a b c Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
- ^ Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
- ^ Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.
External links
- Weisstein, Eric W., "Augmented truncated tetrahedron" ("Johnson solid") at MathWorld.
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- elongated triangular pyramid
- elongated square pyramid
- elongated pentagonal pyramid
- gyroelongated square pyramid
- gyroelongated pentagonal pyramid
- triangular bipyramid
- pentagonal bipyramid
- elongated triangular bipyramid
- elongated square bipyramid
- elongated pentagonal bipyramid
- gyroelongated square bipyramid
- elongated triangular cupola
- elongated square cupola
- elongated pentagonal cupola
- elongated pentagonal rotunda
- gyroelongated triangular cupola
- gyroelongated square cupola
- gyroelongated pentagonal cupola
- gyroelongated pentagonal rotunda
- gyrobifastigium
- triangular orthobicupola
- square orthobicupola
- square gyrobicupola
- pentagonal orthobicupola
- pentagonal gyrobicupola
- pentagonal orthocupolarotunda
- pentagonal gyrocupolarotunda
- pentagonal orthobirotunda
- elongated triangular orthobicupola
- elongated triangular gyrobicupola
- elongated square gyrobicupola
- elongated pentagonal orthobicupola
- elongated pentagonal gyrobicupola
- elongated pentagonal orthocupolarotunda
- elongated pentagonal gyrocupolarotunda
- elongated pentagonal orthobirotunda
- elongated pentagonal gyrobirotunda
- gyroelongated triangular bicupola
- gyroelongated square bicupola
- gyroelongated pentagonal bicupola
- gyroelongated pentagonal cupolarotunda
- gyroelongated pentagonal birotunda
- augmented truncated tetrahedron
- augmented truncated cube
- biaugmented truncated cube
- augmented truncated dodecahedron
- parabiaugmented truncated dodecahedron
- metabiaugmented truncated dodecahedron
- triaugmented truncated dodecahedron
- gyrate rhombicosidodecahedron
- parabigyrate rhombicosidodecahedron
- metabigyrate rhombicosidodecahedron
- trigyrate rhombicosidodecahedron
- diminished rhombicosidodecahedron
- paragyrate diminished rhombicosidodecahedron
- metagyrate diminished rhombicosidodecahedron
- bigyrate diminished rhombicosidodecahedron
- parabidiminished rhombicosidodecahedron
- metabidiminished rhombicosidodecahedron
- gyrate bidiminished rhombicosidodecahedron
- tridiminished rhombicosidodecahedron
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