Deltahedron

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Not to be confused with Deltohedron.
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.

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 eight convex deltahedra

There are eight strictly-convex deltahedra: three are regular polyhedra and Platonic solids, all eight are Johnson 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]
Irregular 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 some of the angles of a cube are relaxed, 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:

triakis tetrahedron tetrakis hexahedron triakis octahedron
(stella octangula) 		pentakis dodecahedron triakis icosahedron
12 triangles 24 triangles 60 triangles

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

Retrieved from "https://mdwiki.org/wiki/Deltahedron"