Deltahedron

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Triaugmented_triangular_prism.png
The triaugmented triangular prism, a convex deltahedron

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 four, six, eight, ten, twelve, fourteen, sixteen, and twenty faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.

Name FacesEdgesVertices
regular tetrahedron464
triangular dipyramid695
regular octahedron8126
pentagonal dipyramid10157
snub disphenoid12188
triaugmented triangular prism14219
gyroelongated square dipyramid162410
regular icosahedron203012

Only three of the deltahedra are Platonic solids (polyhedra in which the number of faces meeting at each vertex is constant). 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.

The deltahedra should not be confused with the deltohedra (spelled with an "o"), polyhedra whose faces are geometric kites.it:Deltaedro

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