Deltahedron


In geometry, a deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta, 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. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.

The eight convex deltahedra

There are only eight strictly-convex deltahedra: three are regular polyhedra, and five are Johnson solids.
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. 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,...
Some smaller examples include:
ImageNameFacesEdgesVerticesVertex configurationsSymmetry group
Augmented octahedron
Augmentation
1 tet + 1 oct		10 1571 × 33
3 × 34
3 × 35
0 × 36		C3v,
Augmented octahedron
Augmentation
1 tet + 1 oct		4
3 		1271 × 33
3 × 34
3 × 35
0 × 36		C3v,
Trigonal trapezohedron
Augmentation
2 tets + 1 oct		12 1882 × 33
0 × 34
6 × 35
0 × 36		C3v,
Trigonal trapezohedron
Augmentation
2 tets + 1 oct		6 1282 × 33
0 × 34
6 × 35
0 × 36		C3v,
Augmentation
2 tets + 1 oct		12 1882 × 33
1 × 34
4 × 35
1 × 36		C2v,
Augmentation
2 tets + 1 oct		2
2
2 		1172 × 33
1 × 34
4 × 35
1 × 36		C2v,
Triangular frustum
Augmentation
3 tets + 1 oct		14 2193 × 33
0 × 34
3 × 35
3 × 36		C3v,
Triangular frustum
Augmentation
3 tets + 1 oct		1
3
1 		963 × 33
0 × 34
3 × 35
3 × 36		C3v,
Elongated octahedron
Augmentation
2 tets + 2 octs		16 24100 × 33
4 × 34
4 × 35
2 × 36		D2h,
Elongated octahedron
Augmentation
2 tets + 2 octs		4
4 		1260 × 33
4 × 34
4 × 35
2 × 36		D2h,
Tetrahedron
Augmentation
4 tets + 1 oct		16 24104 × 33
0 × 34
0 × 35
6 × 36		Td,
Tetrahedron
Augmentation
4 tets + 1 oct		4 644 × 33
0 × 34
0 × 35
6 × 36		Td,
Augmentation
3 tets + 2 octs		18 27111 × 33
2 × 34
5 × 35
3 × 36		D2h,
Augmentation
3 tets + 2 octs		2
1
2
2 		1491 × 33
2 × 34
5 × 35
3 × 36		D2h,
Edge-contracted icosahedron18 27110 × 33
2 × 34
8 × 35
1 × 36		C2v,
Edge-contracted icosahedron12
2 		22100 × 33
2 × 34
8 × 35
1 × 36		C2v,
Triangular bifrustum
Augmentation
6 tets + 2 octs		20 30120 × 33
3 × 34
6 × 35
3 × 36		D3h,
Triangular bifrustum
Augmentation
6 tets + 2 octs		2
6 		1590 × 33
3 × 34
6 × 35
3 × 36		D3h,
triangular cupola
Augmentation
4 tets + 3 octs		22 33130 × 33
3 × 34
6 × 35
4 × 36		C3v,
triangular cupola
Augmentation
4 tets + 3 octs		3
3
1
1 		1590 × 33
3 × 34
6 × 35
4 × 36		C3v,
Triangular bipyramid
Augmentation
8 tets + 2 octs		24 36142 × 33
3 × 34
0 × 35
9 × 36		D3h,
Triangular bipyramid
Augmentation
8 tets + 2 octs		6 952 × 33
3 × 34
0 × 35
9 × 36		D3h,
Hexagonal antiprism24 36140 × 33
0 × 34
12 × 35
2 × 36		D6d,
Hexagonal antiprism12
2 		24120 × 33
0 × 34
12 × 35
2 × 36		D6d,
Truncated tetrahedron
Augmentation
6 tets + 4 octs		28 42160 × 33
0 × 34
12 × 35
4 × 36		Td,
Truncated tetrahedron
Augmentation
6 tets + 4 octs		4
4 		18120 × 33
0 × 34
12 × 35
4 × 36		Td,
Tetrakis cuboctahedron
Octahedron
Augmentation
8 tets + 6 octs		32 48180 × 33
12 × 34
0 × 35
6 × 36		Oh,
Tetrakis cuboctahedron
Octahedron
Augmentation
8 tets + 6 octs		8 1260 × 33
12 × 34
0 × 35
6 × 36		Oh,

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:
Other augmentations of the tetrahedron include:
8 triangles10 triangles12 triangles

Also by adding inverted pyramids to faces:

Excavated dodecahedron
A toroidal deltahedron
60 triangles48 triangles