Deltahedron

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.

A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule 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.(Freudenthal 1947) The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.

The deltahedra should not be confused with the deltohedra (spelled with an "o"), polyhedra whose faces are geometric kites.

The eight convex deltahedra

There are only 8 strictly-convex deltahedra, three are regular polyhedra, and five 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]
Johnson deltahedra
Image	Name	Faces	Edges	Vertices	Vertex configurations	Symmetry group
	triangular dipyramid	6	9	5	2 × 33 3 × 34	D3h, [3,2]
	pentagonal dipyramid	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 dipyramid	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, as its triangular faces would have to meet 6 at a vertex, making some triangles coplanar; however, such a polyhedron can exist with irregular triangles (see octadecahedron).

Non-strictly convex cases

There are infinitely many cases with coplanar triangles, allowing for sections of the infinite triangular tilings. The coplanar triangular faces can be merged into rhombic, trapezoidal, hexagonal, or other equilateral polygon faces.^[1] If the sets of coplanar triangles are considered a single face (called a triamond^[2]), a smaller set of faces, edges, and vertices can be counted. Triamond faces used must be convex, including: , , , , , , and , ...

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 + 4 octs	32	24	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:

• Great icosahedron - a Kepler-Poinsot solid.

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

• Equilateral triakis tetrahedron
• Equilateral tetrakis hexahedron
• Equilateral triakis octahedron (stella octangula)
• Equilateral pentakis dodecahedron
• Equilateral triakis icosahedron

Also by adding inverted pyramids to faces:

• Excavated dodecahedron

Great icosahedron (20 intersecting triangles)

Stella octangula (24 triangles)

Excavated dodecahedron (60 triangles)

A toroidal deltahedron (48 triangles)

External links

• Weisstein, Eric W., "Deltahedron", MathWorld.
• The eight convex deltahedra
• Deltahedron
• Deltahedron

References

1. ↑ The Convex Deltahedra And the Allowance of Coplanar Faces
2. ↑ http://www.interocitors.com/polyhedra/Triamonds/

• 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. )
• H. Martyn Cundy Deltahedra. Math. Gaz. 36, 263-266, Dec 1952.
• H. Martyn Cundy and A. Rollett Deltahedra. §3.11 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 142–144, 1989.
• Charles W. Trigg An Infinite Class of Deltahedra, Mathematics Magazine, Vol. 51, No. 1 (Jan., 1978), pp. 55–57
• M. Gardner Fractal Music, Hypercards, and More: Mathematical Recreations, Scientific American Magazine. New York: W. H. Freeman, pp. 40, 53, and 58-60, 1992.
• Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.  pp. 35–36