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 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.

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

Name	Image	Faces	Edges	Vertices	Vertex configurations
regular tetrahedron		4	6	4	4 × 33
triangular dipyramid		6	9	5	2 × 33 3 × 34
regular octahedron		8	12	6	6 × 34
pentagonal dipyramid		10	15	7	5 × 34 2 × 35
snub disphenoid		12	18	8	4 × 34 4 × 35
triaugmented triangular prism		14	21	9	3 × 34 6 × 35
gyroelongated square dipyramid		16	24	10	2 × 34 8 × 35
regular icosahedron		20	30	12	12 × 35

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.

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.

[edit] Non-convex forms

Some examples of non-convex deltahedra:

1. Great icosahedron - a Kepler-Poinsot solid.
2. stella octangula (simplest of the five regular polyhedral compounds)
3. Wenninger's third stellated icosahedra
4. Equilateral Triakis icosahedron

[edit] External links

• MathWorld
• Eight convex deltahedron
• Deltahedron
• [1]

[edit] References

• H. Martyn Cundy Deltahedra. Math. Gaz. 36, 263-266, Dec 1952. [2]
• H. Martyn Cundy and A. Rollett Deltahedra. §3.11 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 142-144, 1989.
• Charles W. Trigg An Infinite Class of Deltahedra, Mathematics Magazine, Vol. 51, No. 1 (Jan., 1978), pp. 55-57 [3]
• M. Gardner Fractal Music, Hypercards, and More: Mathematical Recreations, Scientific American Magazine. New * York: W. H. Freeman, pp. 40, 53, and 58-60, 1992.
• A. Pugh Polyhedra: A Visual Approach. Berkeley, CA: University of California Press, pp. 35-36, 1976.