A deltahedron is a polyhedron where all the faces are equilateral triangles The name arises from the fact that an equilateral triangle is the shape of the Greek majuscule delta (Δ). Whilst there are infinitely many deltahedra, there are only eight convex deltahedra, which have 4, 6, 8, 10, 12, 14, 16, and 20 faces. Here are the number of faces, edges, and vertices of each deltahedron.
Only three of the deltahedra are platonic solids (where the number of faces that meet at each vertex is constant). These are:
- the 4-face deltahedron (or tetrahedron), in which 3 faces meet at each vertex
- the 8-face deltahedron (or octahedron), in which 4 faces meet at each vertex
- the 20-face deltahedron (or icosahedron), in which 5 faces meet at each vertex.
In the 6-face deltahedron, some vertices have 3 and some 4. In the 10-, 12-, 14-, and 16-face deltahedra, some have 4 and some 5. These five irregular deltahedra belong to the class of Johnson solids, convex polyhedra with regular polygons for faces.
A deltahedron should not be confused with a Deltohedron (spelled with an "o"), also known as a Trapezohedron.