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Deltahedron

March 12, 2023

Not to be confused with Deltohedron.

In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra, all having an even number of faces by the handshaking lemma. Of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces.[1] The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.

The largest strictly-convex deltahedron is the regular icosahedron
This is a truncated tetrahedron with hexagons subdivided into triangles. This figure is not a strictly-convex deltahedron since coplanar faces are not allowed within the definition.

The eight convex deltahedra

There are only eight strictly-convex deltahedra: three are regular polyhedra, and five are Johnson solids. The three regular convex polyhedra are indeed Platonic solids.

Regular deltahedra
Image Name Faces Edges Vertices Vertex configurations Symmetry group
  tetrahedron 4 6 4 4 × 33 Td, [3,3]
  octahedron 8 12 6 6 × 34 Oh, [4,3]
  icosahedron 20 30 12 12 × 35 Ih, [5,3]
Johnson deltahedra
Image Name Faces Edges Vertices Vertex configurations Symmetry group
  triangular bipyramid 6 9 5 2 × 33
3 × 34		 D3h, [3,2]
  pentagonal bipyramid 10 15 7 5 × 34
2 × 35		 D5h, [5,2]
  snub disphenoid 12 18 8 4 × 34
4 × 35		 D2d, [2,2]
  triaugmented triangular prism 14 21 9 3 × 34
6 × 35		 D3h, [3,2]
  gyroelongated square bipyramid 16 24 10 2 × 34
8 × 35		 D4d, [4,2]

In the 6-faced deltahedron, some vertices have degree 3 and some degree 4. In the 10-, 12-, 14-, and 16-faced deltahedra, some vertices have degree 4 and some degree 5. These five irregular deltahedra belong to the class of Johnson solids: convex polyhedra with regular polygons for faces.

Deltahedra retain their shape even if the edges are free to rotate around their vertices so that the angles between edges are fluid. Not all polyhedra have this property: for example, if you relax some of the angles of a cube, the cube can be deformed into a non-right square prism.

There is no 18-faced convex deltahedron.[2] However, the edge-contracted icosahedron gives an example of an octadecahedron that can either be made convex with 18 irregular triangular faces, or made with equilateral triangles that include two coplanar sets of three triangles.

Non-strictly convex cases

There are infinitely many cases with coplanar triangles, allowing for sections of the infinite triangular tilings. If the sets of coplanar triangles are considered a single face, a smaller set of faces, edges, and vertices can be counted. The coplanar triangular faces can be merged into rhombic, trapezoidal, hexagonal, or other equilateral polygon faces. Each face must be a convex polyiamond such as  ,  ,  ,  ,  ,  ,   and  , ...[3]

Some smaller examples include:

Coplanar deltahedra
Image Name Faces Edges Vertices Vertex configurations Symmetry group
  Augmented octahedron
Augmentation
1 tet + 1 oct 		10   15 7 1 × 33
3 × 34
3 × 35
0 × 36		 C3v, [3]
4  
3  		 12
  Trigonal trapezohedron
Augmentation
2 tets + 1 oct 		12   18 8 2 × 33
0 × 34
6 × 35
0 × 36		 C3v, [3]
6   12
  Augmentation
2 tets + 1 oct 		12   18 8 2 × 33
1 × 34
4 × 35
1 × 36		 C2v, [2]
2  
2  
2  		 11 7
  Triangular frustum
Augmentation
3 tets + 1 oct 		14   21 9 3 × 33
0 × 34
3 × 35
3 × 36		 C3v, [3]
1  
3  
1  		 9 6
  Elongated octahedron
Augmentation
2 tets + 2 octs 		16   24 10 0 × 33
4 × 34
4 × 35
2 × 36		 D2h, [2,2]
4  
4  		 12 6
  Tetrahedron
Augmentation
4 tets + 1 oct 		16   24 10 4 × 33
0 × 34
0 × 35
6 × 36		 Td, [3,3]
4   6 4
  Augmentation
3 tets + 2 octs 		18   27 11 1 × 33
2 × 34
5 × 35
3 × 36		 D2h, [2,2]
2  
1  
2  
2  		 14 9
  Edge-contracted icosahedron 18   27 11 0 × 33
2 × 34
8 × 35
1 × 36		 C2v, [2]
12  
2  		 22 10
  Triangular bifrustum
Augmentation
6 tets + 2 octs 		20   30 12 0 × 33
3 × 34
6 × 35
3 × 36		 D3h, [3,2]
2  
6  		 15 9
  triangular cupola
Augmentation
4 tets + 3 octs 		22   33 13 0 × 33
3 × 34
6 × 35
4 × 36		 C3v, [3]
3  
3  
1  
1  		 15 9
  Triangular bipyramid
Augmentation
8 tets + 2 octs 		24   36 14 2 × 33
3 × 34
0 × 35
9 × 36		 D3h, [3]
6   9 5
  Hexagonal antiprism 24   36 14 0 × 33
0 × 34
12 × 35
2 × 36		 D6d, [12,2+]
12  
2  		 24 12
  Truncated tetrahedron
Augmentation
6 tets + 4 octs 		28   42 16 0 × 33
0 × 34
12 × 35
4 × 36		 Td, [3,3]
4  
4  		 18 12
  Tetrakis cuboctahedron
Octahedron
Augmentation
8 tets + 6 octs 		32   48 18 0 × 33
12 × 34
0 × 35
6 × 36		 Oh, [4,3]
8   12 6

Non-convex forms

There are an infinite number of nonconvex forms.

Some examples of face-intersecting deltahedra:

Other nonconvex deltahedra can be generated by adding equilateral pyramids to the faces of all 5 Platonic solids:

Other augmentations of the tetrahedron include:

Augmented tetrahedra
     
8 triangles 10 triangles 12 triangles

Also by adding inverted pyramids to faces:

 
Excavated dodecahedron 		 
A toroidal deltahedron
60 triangles 48 triangles

See also

References

  1. ^ Freudenthal, H; van der Waerden, B. L. (1947), "Over een bewering van Euclides ("On an Assertion of Euclid")", Simon Stevin (in Dutch), 25: 115–128 (They showed that there are just 8 convex deltahedra. )
  2. ^ Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.1080/0025570X.1978.11976675, JSTOR 2689647.
  3. ^ The Convex Deltahedra And the Allowance of Coplanar Faces

Further reading

  • Rausenberger, O. (1915), "Konvexe pseudoreguläre Polyeder", Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 46: 135–142.
  • Cundy, H. Martyn (December 1952), "Deltahedra", Mathematical Gazette, 36: 263–266, doi:10.2307/3608204, JSTOR 3608204.
  • Cundy, H. Martyn; Rollett, A. (1989), "3.11. Deltahedra", Mathematical Models (3rd ed.), Stradbroke, England: Tarquin Pub., pp. 142–144.
  • Gardner, Martin (1992), Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American, New York: W. H. Freeman, pp. 40, 53, and 58-60.
  • Pugh, Anthony (1976), Polyhedra: A visual approach, California: University of California Press Berkeley, ISBN 0-520-03056-7 pp. 35–36

External links

March 12, 2023
deltahedron, confused, with, deltohedron, geometry, deltahedron, plural, deltahedra, polyhedron, whose, faces, equilateral, triangles, name, taken, from, greek, upper, case, delta, which, shape, equilateral, triangle, there, infinitely, many, deltahedra, havin. Not to be confused with Deltohedron In geometry a deltahedron plural deltahedra is a polyhedron whose faces are all equilateral triangles The name is taken from the Greek upper case delta D which has the shape of an equilateral triangle There are infinitely many deltahedra all having an even number of faces by the handshaking lemma Of these only eight are convex having 4 6 8 10 12 14 16 and 20 faces 1 The number of faces edges and vertices is listed below for each of the eight convex deltahedra The largest strictly convex deltahedron is the regular icosahedron This is a truncated tetrahedron with hexagons subdivided into triangles This figure is not a strictly convex deltahedron since coplanar faces are not allowed within the definition Contents 1 The eight convex deltahedra 2 Non strictly convex cases 3 Non convex forms 4 See also 5 References 6 Further reading 7 External linksThe eight convex deltahedra EditThere are only eight strictly convex deltahedra three are regular polyhedra and five are Johnson solids The three regular convex polyhedra are indeed Platonic solids Regular deltahedraImage Name Faces Edges Vertices Vertex configurations Symmetry group tetrahedron 4 6 4 4 33 Td 3 3 octahedron 8 12 6 6 34 Oh 4 3 icosahedron 20 30 12 12 35 Ih 5 3 Johnson deltahedraImage Name Faces Edges Vertices Vertex configurations Symmetry group triangular bipyramid 6 9 5 2 333 34 D3h 3 2 pentagonal bipyramid 10 15 7 5 342 35 D5h 5 2 snub disphenoid 12 18 8 4 344 35 D2d 2 2 triaugmented triangular prism 14 21 9 3 346 35 D3h 3 2 gyroelongated square bipyramid 16 24 10 2 348 35 D4d 4 2 In the 6 faced deltahedron some vertices have degree 3 and some degree 4 In the 10 12 14 and 16 faced deltahedra some vertices have degree 4 and some degree 5 These five irregular deltahedra belong to the class of Johnson solids convex polyhedra with regular polygons for faces Deltahedra retain their shape even if the edges are free to rotate around their vertices so that the angles between edges are fluid Not all polyhedra have this property for example if you relax some of the angles of a cube the cube can be deformed into a non right square prism There is no 18 faced convex deltahedron 2 However the edge contracted icosahedron gives an example of an octadecahedron that can either be made convex with 18 irregular triangular faces or made with equilateral triangles that include two coplanar sets of three triangles Non strictly convex cases EditThere are infinitely many cases with coplanar triangles allowing for sections of the infinite triangular tilings If the sets of coplanar triangles are considered a single face a smaller set of faces edges and vertices can be counted The coplanar triangular faces can be merged into rhombic trapezoidal hexagonal or other equilateral polygon faces Each face must be a convex polyiamond such as and 3 Some smaller examples include Coplanar deltahedra Image Name Faces Edges Vertices Vertex configurations Symmetry group Augmented octahedronAugmentation1 tet 1 oct 10 15 7 1 333 343 350 36 C3v 3 4 3 12 Trigonal trapezohedronAugmentation2 tets 1 oct 12 18 8 2 330 346 350 36 C3v 3 6 12 Augmentation2 tets 1 oct 12 18 8 2 331 344 351 36 C2v 2 2 2 2 11 7 Triangular frustumAugmentation3 tets 1 oct 14 21 9 3 330 343 353 36 C3v 3 1 3 1 9 6 Elongated octahedronAugmentation2 tets 2 octs 16 24 10 0 334 344 352 36 D2h 2 2 4 4 12 6 TetrahedronAugmentation4 tets 1 oct 16 24 10 4 330 340 356 36 Td 3 3 4 6 4 Augmentation3 tets 2 octs 18 27 11 1 332 345 353 36 D2h 2 2 2 1 2 2 14 9 Edge contracted icosahedron 18 27 11 0 332 348 351 36 C2v 2 12 2 22 10 Triangular bifrustumAugmentation6 tets 2 octs 20 30 12 0 333 346 353 36 D3h 3 2 2 6 15 9 triangular cupolaAugmentation4 tets 3 octs 22 33 13 0 333 346 354 36 C3v 3 3 3 1 1 15 9 Triangular bipyramidAugmentation8 tets 2 octs 24 36 14 2 333 340 359 36 D3h 3 6 9 5 Hexagonal antiprism 24 36 14 0 330 3412 352 36 D6d 12 2 12 2 24 12 Truncated tetrahedronAugmentation6 tets 4 octs 28 42 16 0 330 3412 354 36 Td 3 3 4 4 18 12 Tetrakis cuboctahedronOctahedronAugmentation8 tets 6 octs 32 48 18 0 3312 340 356 36 Oh 4 3 8 12 6Non convex forms EditThere are an infinite number of nonconvex forms Some examples of face intersecting deltahedra Great icosahedron a Kepler Poinsot solid with 20 intersecting triangles Other nonconvex deltahedra can be generated by adding equilateral pyramids to the faces of all 5 Platonic solids triakis tetrahedron tetrakis hexahedron triakis octahedron stella octangula pentakis dodecahedron triakis icosahedron12 triangles 24 triangles 60 trianglesOther augmentations of the tetrahedron include Augmented tetrahedra 8 triangles 10 triangles 12 trianglesAlso by adding inverted pyramids to faces Excavated dodecahedron Excavated dodecahedron A toroidal deltahedron60 triangles 48 trianglesSee also EditSimplicial polytope polytopes with all simplex facetsReferences Edit Freudenthal H van der Waerden B L 1947 Over een bewering van Euclides On an Assertion of Euclid Simon Stevin in Dutch 25 115 128 They showed that there are just 8 convex deltahedra Trigg Charles W 1978 An Infinite Class of Deltahedra Mathematics Magazine 51 1 55 57 doi 10 1080 0025570X 1978 11976675 JSTOR 2689647 The Convex Deltahedra And the Allowance of Coplanar FacesFurther reading EditRausenberger O 1915 Konvexe pseudoregulare Polyeder Zeitschrift fur mathematischen und naturwissenschaftlichen Unterricht 46 135 142 Cundy H Martyn December 1952 Deltahedra Mathematical Gazette 36 263 266 doi 10 2307 3608204 JSTOR 3608204 Cundy H Martyn Rollett A 1989 3 11 Deltahedra Mathematical Models 3rd ed Stradbroke England Tarquin Pub pp 142 144 Gardner Martin 1992 Fractal Music Hypercards and More Mathematical Recreations from Scientific American New York W H Freeman pp 40 53 and 58 60 Pugh Anthony 1976 Polyhedra A visual approach California University of California Press Berkeley ISBN 0 520 03056 7 pp 35 36External links EditWeisstein Eric W Deltahedron MathWorld The eight convex deltahedra Deltahedron Deltahedron Retrieved from https en wikipedia org w index php title Deltahedron amp oldid 1095008195, wikipedia, wiki, book, books, library,

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