# Deltahedron

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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. 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 Edit

Name Image Faces Edges Vertices Vertex configurations Symmetry group
regular tetrahedron 60px 4 6 4 4 × 33 $T_d$
triangular dipyramid 60px 6 9 5 2 × 33
3 × 34
$D_{3h}$
regular octahedron 60px 8 12 6 6 × 34 $O_h$
pentagonal dipyramid 60px 10 15 7 5 × 34
2 × 35
$D_{5h}$
snub disphenoid 60px 12 18 8 4 × 34
4 × 35
$D_{2d}$
triaugmented triangular prism 60px 14 21 9 3 × 34
6 × 35
$D_{3h}$
gyroelongated square dipyramid 60px 16 24 10 2 × 34
8 × 35
$D_{4d}$
regular icosahedron 60px 20 30 12 12 × 35 $I_h$

Three of the deltahedra are Platonic solids (polyhedra in which a constant number of identical regular faces meet at each vertex). These are:

• the 4-faced deltahedron (or tetrahedron), in which three faces meet at each vertex
• the 8-faced deltahedron (or octahedron), in which four faces meet at each vertex
• the 20-faced deltahedron (or icosahedron), in which five faces meet at each vertex

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.

## Non-convex forms Edit

There are an infinite number of nonconvex forms.

Some examples of non-convex deltahedra:

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

1. Equilateral triakis tetrahedron
2. Equilateral tetrakis hexahedron
3. Equilateral triakis octahedron (stella octangula)
4. Equilateral pentakis dodecahedron
5. Equilateral triakis icosahedron

Also by adding inverted pyramids to faces:

 160pxGreat icosahedron(20 intersecting triangles) 160pxstella octangula(24 triangles) 160pxThird stellation of icosahedron(60 triangles) 160pxA toroidal deltahedron(48 triangles)