As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four.
The following formulae for the height (), surface area () and volume () can be used if all faces are regular, with edge length :[2]
Dual polyhedronedit
The dual polyhedron of the triangular bipyramid is the triangular prism, with five faces: two parallel equilateral triangles linked by a chain of three rectangles. Although the triangular prism has a form that is a uniform polyhedron (with square faces), the dual of the Johnson solid form of the bipyramid has rectangular rather than square faces, and is not uniform.
Triangular prism
Net
Related polyhedra and honeycombsedit
The triangular bipyramid, dt{2,3}, can be in sequence rectified, rdt{2,3}, truncated, trdt{2,3} and alternated (snubbed), srdt{2,3}:
The triangular bipyramid can be constructed by augmentation of smaller ones, specifically two stacked regular octahedra with 3 triangular bipyramids added around the sides, and 1 tetrahedron above and below. This polyhedron has 24 equilateral triangle faces, but it is not a Johnson solid because it has coplanar faces. It is a coplanar 24-triangle deltahedron. This polyhedron exists as the augmentation of cells in a gyrated alternated cubic honeycomb. Larger triangular polyhedra can be generated similarly, like 9, 16 or 25 triangles per larger triangle face, seen as a section of a triangular tiling.
When projected onto a sphere, it resembles a compound of a trigonal hosohedron and trigonal dihedron. It is part of an infinite series of dual pair compounds of regular polyhedra projected onto spheres. The triangular bipyramid can be referred to as a deltoidal hexahedron for consistency with the other solids in the series, although the "deltoids" are triangles instead of kites in this case, as the angle from the dihedron is 180 degrees.
*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
triangular, bipyramid, related, molecular, geometrical, structure, trigonal, bipyramid, molecular, geometry, geometry, triangular, bipyramid, dipyramid, type, hexahedron, being, first, infinite, face, transitive, bipyramids, dual, triangular, prism, with, isos. For the related molecular geometrical structure see Trigonal bipyramid molecular geometry In geometry the triangular bipyramid or dipyramid is a type of hexahedron being the first in the infinite set of face transitive bipyramids It is the dual of the triangular prism with 6 isosceles triangle faces Triangular bipyramidTypeBipyramid JohnsonJ11 J12 J13Faces6 trianglesEdges9Vertices5Schlafli symbol 3 Coxeter diagramSymmetry groupD3h 3 2 223 order 12Rotation groupD3 3 2 223 order 6Dual polyhedronTriangular prismFace configurationV3 4 4PropertiesConvex face transitive3D model of a triangular bipyramidNetAs the name suggests it can be constructed by joining two tetrahedra along one face Although all its faces are congruent and the solid is face transitive it is not a Platonic solid because some vertices adjoin three faces and others adjoin four The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids J12 A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra that is they are not Platonic solids Archimedean solids prisms or antiprisms They were named by Norman Johnson who first listed these polyhedra in 1966 1 As a Johnson solid with all faces equilateral triangles it is also a deltahedron Contents 1 Formulae 2 Dual polyhedron 3 Related polyhedra and honeycombs 4 See also 5 References 6 External linksFormulae editThe following formulae for the height H displaystyle H nbsp surface area A displaystyle A nbsp and volume V displaystyle V nbsp can be used if all faces are regular with edge length L displaystyle L nbsp 2 H L 2 6 3 L 1 632993162 displaystyle H L cdot frac 2 sqrt 6 3 approx L cdot 1 632993162 nbsp A L 2 3 3 2 L 2 2 598076211 displaystyle A L 2 cdot frac 3 sqrt 3 2 approx L 2 cdot 2 598076211 nbsp V L 3 2 6 L 3 0 235702260 displaystyle V L 3 cdot frac sqrt 2 6 approx L 3 cdot 0 235702260 nbsp Dual polyhedron editThe dual polyhedron of the triangular bipyramid is the triangular prism with five faces two parallel equilateral triangles linked by a chain of three rectangles Although the triangular prism has a form that is a uniform polyhedron with square faces the dual of the Johnson solid form of the bipyramid has rectangular rather than square faces and is not uniform Triangular prism Net nbsp nbsp Related polyhedra and honeycombs editThe triangular bipyramid dt 2 3 can be in sequence rectified rdt 2 3 truncated trdt 2 3 and alternated snubbed srdt 2 3 nbsp The triangular bipyramid can be constructed by augmentation of smaller ones specifically two stacked regular octahedra with 3 triangular bipyramids added around the sides and 1 tetrahedron above and below This polyhedron has 24 equilateral triangle faces but it is not a Johnson solid because it has coplanar faces It is a coplanar 24 triangle deltahedron This polyhedron exists as the augmentation of cells in a gyrated alternated cubic honeycomb Larger triangular polyhedra can be generated similarly like 9 16 or 25 triangles per larger triangle face seen as a section of a triangular tiling nbsp The triangular bipyramid can form a tessellation of space with octahedra or with truncated tetrahedra 3 nbsp Layers of the uniform quarter cubic honeycomb can be shifted to pair up regular tetrahedral cells which combined into triangular bipyramids nbsp The gyrated tetrahedral octahedral honeycomb has pairs of adjacent regular tetrahedra that can be seen as triangular bipyramids When projected onto a sphere it resembles a compound of a trigonal hosohedron and trigonal dihedron It is part of an infinite series of dual pair compounds of regular polyhedra projected onto spheres The triangular bipyramid can be referred to as a deltoidal hexahedron for consistency with the other solids in the series although the deltoids are triangles instead of kites in this case as the angle from the dihedron is 180 degrees n32 symmetry mutation of dual expanded tilings V3 4 n 4 Symmetry n32 n 3 Spherical Euclid Compact hyperb Paraco 232 2 3 332 3 3 432 4 3 532 5 3 632 6 3 732 7 3 832 8 3 32 3 FigureConfig nbsp V3 4 2 4 nbsp V3 4 3 4 nbsp V3 4 4 4 nbsp V3 4 5 4 nbsp V3 4 6 4 nbsp V3 4 7 4 nbsp V3 4 8 4 nbsp V3 4 4See also editTrigonal bipyramidal molecular geometry Boerdijk Coxeter helix an extension of the triangular bipyramid by adding more tetrahedrons Regular right symmetric n gonal bipyramids Bipyramid name Digonal bipyramid Triangular bipyramid See J12 Square bipyramid See O Pentagonal bipyramid See J13 Hexagonal bipyramid Heptagonal bipyramid Octagonal bipyramid Enneagonal bipyramid Decagonal bipyramid Apeirogonal bipyramidPolyhedron image nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Spherical tiling image nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Plane tiling image nbsp Face config V2 4 4 V3 4 4 V4 4 4 V5 4 4 V6 4 4 V7 4 4 V8 4 4 V9 4 4 V10 4 4 V 4 4Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp References edit 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 Zbl 0132 14603 Sapina R Area and volume of the Johnson solid J Problemas y Ecuaciones in Spanish ISSN 2659 9899 Retrieved 2020 09 01 J12 honeycomb External links editWeisstein Eric W Triangular dipyramid Johnson solid at MathWorld Conway Notation for Polyhedra Try dP3 Retrieved from https en wikipedia org w index php title Triangular bipyramid amp oldid 1153102137, wikipedia, wiki, book, books, library,
