Deltahedron

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Image:Triaugmented triangular prism.png

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	Faces	Edges	Vertices
regular tetrahedron	4	6	4
triangular dipyramid	6	9	5
regular octahedron	8	12	6
pentagonal dipyramid	10	15	7
snub disphenoid	12	18	8
triaugmented triangular prism	14	21	9
gyroelongated square dipyramid	16	24	10
regular icosahedron	20	30	12

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

There is one non-convex uniform polyhedron made of all triangles, the great icosahedron.

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.

An example of a non-convex deltahedron is the simplest of the five regular polyhedral compounds: the stella octangula. One of the stellated icosahedra (I₂₀) is also a (non-convex) deltahedron.[1]

The deltahedra should not be confused with the deltohedra (spelled with an "o"), polyhedra whose faces are geometric kites.de:Deltaeder it:Deltaedro ja:デルタ多面体