Deltahedron

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, but 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.

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.

The eight convex deltahedra

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

Regular deltahedra
ImageNameFacesEdgesVerticesVertex configurationsSymmetry group
tetrahedron4644 × 33Td, [3,3]
octahedron81266 × 34Oh, [4,3]
icosahedron20301212 × 35Ih, [5,3]
Johnson deltahedra
ImageNameFacesEdgesVerticesVertex configurationsSymmetry group
triangular bipyramid6952 × 33
3 × 34		D3h, [3,2]
pentagonal bipyramid101575 × 34
2 × 35		D5h, [5,2]
snub disphenoid121884 × 34
4 × 35		D2d, [2,2]
triaugmented triangular prism142193 × 34
6 × 35		D3h, [3,2]
gyroelongated square bipyramid1624102 × 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
ImageNameFacesEdgesVerticesVertex configurationsSymmetry 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		 12 188 2 × 33
1 × 34
4 × 35
1 × 36		 C2v, [2]
2
2
2 		117
Triangular frustum
Augmentation
3 tets + 1 oct		 14 219 3 × 33
0 × 34
3 × 35
3 × 36		 C3v, [3]
1
3
1 		96
Elongated octahedron
Augmentation
2 tets + 2 octs		 16 2410 0 × 33
4 × 34
4 × 35
2 × 36		 D2h, [2,2]
4
4 		126
Tetrahedron
Augmentation
4 tets + 1 oct		 16 2410 4 × 33
0 × 34
0 × 35
6 × 36		 Td, [3,3]
4 64
Augmentation
3 tets + 2 octs		 18 2711 1 × 33
2 × 34
5 × 35
3 × 36		 D2h, [2,2]
2
1
2
2 		149
Edge-contracted icosahedron 18 2711 0 × 33
2 × 34
8 × 35
1 × 36		 C2v, [2]
12
2 		2210
Triangular bifrustum
Augmentation
6 tets + 2 octs		 20 3012 0 × 33
3 × 34
6 × 35
3 × 36		 D3h, [3,2]
2
6 		159
triangular cupola
Augmentation
4 tets + 3 octs		 22 3313 0 × 33
3 × 34
6 × 35
4 × 36		 C3v, [3]
3
3
1
1 		159
Triangular bipyramid
Augmentation
8 tets + 2 octs		 24 3614 2 × 33
3 × 34
0 × 35
9 × 36		 D3h, [3]
6 95
Hexagonal antiprism 24 3614 0 × 33
0 × 34
12 × 35
2 × 36		 D6d, [12,2+]
12
2 		2412
Truncated tetrahedron
Augmentation
6 tets + 4 octs		 28 4216 0 × 33
0 × 34
12 × 35
4 × 36		 Td, [3,3]
4
4 		1812
Tetrakis cuboctahedron
Octahedron
Augmentation
8 tets + 6 octs		 32 4818 0 × 33
12 × 34
0 × 35
6 × 36		 Oh, [4,3]
8 126

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 regular polyhedra:

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

Other augmentations of the tetrahedron include:

Examples: 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
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