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, 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	T_d, [3,3]
	octahedron	8	12	6	6 × 3⁴	O_h, [4,3]
	icosahedron	20	30	12	12 × 3⁵	I_h, [5,3]
Johnson deltahedra
Image	Name	Faces	Edges	Vertices	Vertex configurations	Symmetry group
	triangular bipyramid	6	9	5	2 × 3³ 3 × 3⁴	D_3h, [3,2]
	pentagonal bipyramid	10	15	7	5 × 3⁴ 2 × 3⁵	D_5h, [5,2]
	snub disphenoid	12	18	8	4 × 3⁴ 4 × 3⁵	D_2d, [2,2]
	triaugmented triangular prism	14	21	9	3 × 3⁴ 6 × 3⁵	D_3h, [3,2]
	gyroelongated square bipyramid	16	24	10	2 × 3⁴ 8 × 3⁵	D_4d, [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
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	C_3v, [3]
Augmented octahedron Augmentation 1 tet + 1 oct	4 3	12	7	1 × 3³ 3 × 3⁴ 3 × 3⁵ 0 × 3⁶	C_3v, [3]
Trigonal trapezohedron Augmentation 2 tets + 1 oct	12	18	8	2 × 3³ 0 × 3⁴ 6 × 3⁵ 0 × 3⁶	C_3v, [3]
Trigonal trapezohedron Augmentation 2 tets + 1 oct	6	12	8	2 × 3³ 0 × 3⁴ 6 × 3⁵ 0 × 3⁶	C_3v, [3]
Augmentation 2 tets + 1 oct	12	18	8	2 × 3³ 1 × 3⁴ 4 × 3⁵ 1 × 3⁶	C_2v, [2]
Augmentation 2 tets + 1 oct	2 2 2	11	7	2 × 3³ 1 × 3⁴ 4 × 3⁵ 1 × 3⁶	C_2v, [2]
Triangular frustum Augmentation 3 tets + 1 oct	14	21	9	3 × 3³ 0 × 3⁴ 3 × 3⁵ 3 × 3⁶	C_3v, [3]
Triangular frustum Augmentation 3 tets + 1 oct	1 3 1	9	6	3 × 3³ 0 × 3⁴ 3 × 3⁵ 3 × 3⁶	C_3v, [3]
Elongated octahedron Augmentation 2 tets + 2 octs	16	24	10	0 × 3³ 4 × 3⁴ 4 × 3⁵ 2 × 3⁶	D_2h, [2,2]
Elongated octahedron Augmentation 2 tets + 2 octs	4 4	12	6	0 × 3³ 4 × 3⁴ 4 × 3⁵ 2 × 3⁶	D_2h, [2,2]
Tetrahedron Augmentation 4 tets + 1 oct	16	24	10	4 × 3³ 0 × 3⁴ 0 × 3⁵ 6 × 3⁶	T_d, [3,3]
Tetrahedron Augmentation 4 tets + 1 oct	4	6	4	4 × 3³ 0 × 3⁴ 0 × 3⁵ 6 × 3⁶	T_d, [3,3]
Augmentation 3 tets + 2 octs	18	27	11	1 × 3³ 2 × 3⁴ 5 × 3⁵ 3 × 3⁶	D_2h, [2,2]
Augmentation 3 tets + 2 octs	2 1 2 2	14	9	1 × 3³ 2 × 3⁴ 5 × 3⁵ 3 × 3⁶	D_2h, [2,2]
Edge-contracted icosahedron	18	27	11	0 × 3³ 2 × 3⁴ 8 × 3⁵ 1 × 3⁶	C_2v, [2]
Edge-contracted icosahedron	12 2	22	10	0 × 3³ 2 × 3⁴ 8 × 3⁵ 1 × 3⁶	C_2v, [2]
Triangular bifrustum Augmentation 6 tets + 2 octs	20	30	12	0 × 3³ 3 × 3⁴ 6 × 3⁵ 3 × 3⁶	D_3h, [3,2]
Triangular bifrustum Augmentation 6 tets + 2 octs	2 6	15	9	0 × 3³ 3 × 3⁴ 6 × 3⁵ 3 × 3⁶	D_3h, [3,2]
triangular cupola Augmentation 4 tets + 3 octs	22	33	13	0 × 3³ 3 × 3⁴ 6 × 3⁵ 4 × 3⁶	C_3v, [3]
triangular cupola Augmentation 4 tets + 3 octs	3 3 1 1	15	9	0 × 3³ 3 × 3⁴ 6 × 3⁵ 4 × 3⁶	C_3v, [3]
Triangular bipyramid Augmentation 8 tets + 2 octs	24	36	14	2 × 3³ 3 × 3⁴ 0 × 3⁵ 9 × 3⁶	D_3h, [3]
Triangular bipyramid Augmentation 8 tets + 2 octs	6	9	5	2 × 3³ 3 × 3⁴ 0 × 3⁵ 9 × 3⁶	D_3h, [3]
Hexagonal antiprism	24	36	14	0 × 3³ 0 × 3⁴ 12 × 3⁵ 2 × 3⁶	D_6d, [12,2⁺]
Hexagonal antiprism	12 2	24	12	0 × 3³ 0 × 3⁴ 12 × 3⁵ 2 × 3⁶	D_6d, [12,2⁺]
Truncated tetrahedron Augmentation 6 tets + 4 octs	28	42	16	0 × 3³ 0 × 3⁴ 12 × 3⁵ 4 × 3⁶	T_d, [3,3]
Truncated tetrahedron Augmentation 6 tets + 4 octs	4 4	18	12	0 × 3³ 0 × 3⁴ 12 × 3⁵ 4 × 3⁶	T_d, [3,3]
Tetrakis cuboctahedron Octahedron Augmentation 8 tets + 6 octs	32	48	18	0 × 3³ 12 × 3⁴ 0 × 3⁵ 6 × 3⁶	O_h, [4,3]
	8	12	6	0 × 3³ 12 × 3⁴ 0 × 3⁵ 6 × 3⁶	O_h, [4,3]

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

60 triangles	48 triangles
Excavated dodecahedron	A toroidal deltahedron

References

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 Faces

External links

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[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