Deltahedron

In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all (congruent) equilateral triangles. The name is taken from the Greek upper case delta letter (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra. By the handshaking lemma, each deltahedron has an even number of faces. Only eight deltahedra are strictly convex; these have 4, 6, 8, 10, 12, 14, 16, or 20 faces.^[1] The eight convex deltahedra, with their respective numbers of faces, edges, and vertices, are listed below.

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

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:

• 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 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

Excavated dodecahedron	A toroidal deltahedron
60 triangles	48 triangles

See also

• Simplicial polytope - polytopes with all simplex facets

References

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

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

This page was last edited on 1 May 2024, at 16:57