Deltahedron Explained

Deltahedron should not be confused with Deltohedron.

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, and 20 faces.^[1] These eight deltahedra, with their respective numbers of faces, edges, and vertices, are listed below.

The eight strictly convex deltahedra

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

Regular strictly convex deltahedra! Image
Name	Faces	Edges	Vertices	Vertex configurations	Symmetry group
		4	6	4	4 × 3³	T_d, [3,3]
		8	12	6	6 × 3⁴	O_h, [4,3]
		20	30	12	12 × 3⁵	I_h, [5,3]

Non-regular strictly convex deltahedra! Image
Name	Faces	Edges	Vertices	Vertex configurations	Symmetry group
		6	9	5	2 × 3³ 3 × 3⁴	D_3h, [3,2]
		10	15	7	5 × 3⁴ 2 × 3⁵	D_5h, [5,2]
		12	18	8	4 × 3⁴ 4 × 3⁵	D_2d, [4,2<sup>+</sup>]
		14	21	9	3 × 3⁴ 6 × 3⁵	D_3h, [3,2]
		16	24	10	2 × 3⁴ 8 × 3⁵	D_4d, [8,2<sup>+</sup>]

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 non-regular deltahedra belong to the class of Johnson solids: non-uniform strictly convex polyhedra with regular polygons for faces.

A deltahedron retains its shape: even if its edges are free to rotate around their vertices (so that the angles between them are fluid), they don't move. Not all polyhedra have this property: for example, if some of the angles of a cube are relaxed, it can be deformed into a non-right square prism or even into a rhombohedron with no right angle at all.

There is no 18-faced strictly convex deltahedron.^[2] However, the edge-contracted icosahedron gives an example of an octadecahedron that can either be made strictly convex with 18 irregular triangular faces, or made equilateral with 18 (regular) triangular faces that include two sets of three coplanar triangles.

Non-strictly convex cases

There are infinitely many cases with coplanar triangles, allowing for convex sections of the infinite triangular tiling. 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 polygonal faces. Each face must be a convex polyiamond, such as,,,,,,,, ...^[3]

Some small examples include:

Deltahedra with coplanar faces! 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	C_3v, [3]
		4 3	12
	Trigonal trapezohedron Augmentation 2 tets + 1 oct	12	18	8	2 × 3³ 0 × 3⁴ 6 × 3⁵ 0 × 3⁶	D_3d, [6,2<sup>+</sup>]
		6	12
	Augmentation 2 tets + 1 oct	12	18	8	2 × 3³ 1 × 3⁴ 4 × 3⁵ 1 × 3⁶	C_2v, [2]
		2 2 2	11	7
	Triangular frustum Augmentation 3 tets + 1 oct	14	21	9	3 × 3³ 0 × 3⁴ 3 × 3⁵ 3 × 3⁶	C_3v, [3]
		1 3 1	9	6
	Elongated octahedron Augmentation 2 tets + 2 octs	16	24	10	0 × 3³ 4 × 3⁴ 4 × 3⁵ 2 × 3⁶	D_2h, [2,2]
		4 4	12	6
	Tetrahedron Augmentation 4 tets + 1 oct	16	24	10	4 × 3³ 0 × 3⁴ 0 × 3⁵ 6 × 3⁶	T_d, [3,3]
		4	6	4
	Augmentation 3 tets + 2 octs	18	27	11	1 × 3³ 2 × 3⁴ 5 × 3⁵ 3 × 3⁶	through a plane
		2 1 2 2	14	9
	Edge-contracted icosahedron	18	27	11	0 × 3³ 2 × 3⁴ 8 × 3⁵ 1 × 3⁶	C_2v, [2]
		12 2	22	10
	Triangular bifrustum Augmentation 6 tets + 2 octs	20	30	12	0 × 3³ 3 × 3⁴ 6 × 3⁵ 3 × 3⁶	D_3h, [3,2]
		2 6	15	9
	Triangular cupola Augmentation 4 tets + 3 octs	22	33	13	0 × 3³ 3 × 3⁴ 6 × 3⁵ 4 × 3⁶	C_3v, [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	D_3h, [3,2]
		6	9	5
	Hexagonal antiprism	24	36	14	0 × 3³ 0 × 3⁴ 12 × 3⁵ 2 × 3⁶	D_6d, [12,2<sup>+</sup>]
		12 2	24	12
	Truncated tetrahedron Augmentation 6 tets + 4 octs	28	42	16	0 × 3³ 0 × 3⁴ 12 × 3⁵ 4 × 3⁶	T_d, [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	O_h, [4,3]
		8	12	6

Non-convex forms

There are an infinite number of non-convex deltahedra.

Five non-convex deltahedra can be generated by adding an equilateral pyramid to every face of a Platonic solid:

Other non-convex deltahedra can be generated by assembling several regular tetrahedra:

Like all toroidal polyhedra, toroidal deltahedra are non-convex; example:

When possible, adding an inverted equilateral pyramid to every face of a polyhedron makes a non-convex deltahedron; example:

Like all self-intersecting polyhedra, self-intersecting deltahedra are non-convex; example:

Great icosahedron — a Kepler-Poinsot solid, with 20 intersecting triangles:

External links

Notes and References

(They showed that there are just eight strictly convex deltahedra.)
.
http://www.interocitors.com/polyhedra/Deltahedra/Convex The Convex Deltahedra And the Allowance of Coplanar Faces