Deltahedron

This is a truncated tetrahedron with hexagons subdivided into triangles. This figure is not a deltahedron since coplanar faces are not allowed within the definition.

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.(Freudenthal 1947) 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

There are only 8 strictly-convex deltahedra:

Name	Image	Faces	Edges	Vertices	Vertex configurations	Symmetry group
regular tetrahedron		4	6	4	4 × 33	Td
triangular dipyramid		6	9	5	2 × 33 3 × 34	D3h
regular octahedron		8	12	6	6 × 34	Oh
pentagonal dipyramid		10	15	7	5 × 34 2 × 35	D5h
snub disphenoid		12	18	8	4 × 34 4 × 35	D2d
triaugmented triangular prism		14	21	9	3 × 34 6 × 35	D3h
gyroelongated square dipyramid		16	24	10	2 × 34 8 × 35	D4d
regular icosahedron		20	30	12	12 × 35	Ih

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

There are an infinite number of nonconvex forms.

Some examples of face-intersecting deltahedra:

• Great icosahedron - a Kepler-Poinsot solid.

Other nonconvex 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:

• Wenninger's third stellated icosahedra

Great icosahedron (20 intersecting triangles)

stella octangula (24 triangles)

Third stellation of icosahedron (60 triangles)

A toroidal deltahedron (48 triangles)

External links

• Weisstein, Eric W., "Deltahedron" from MathWorld.
• Eight convex deltahedron
• Deltahedron
• Deltahedron

References

• Freudenthal, H; van der Waerden, B. L. (1947), "Over een bewering van Euclides ("On an Assertion of Euclid")" (in Dutch), Simon Stevin 25: 115–128  (They showed that there are just 8 convex deltahedra. )
• H. Martyn Cundy Deltahedra. Math. Gaz. 36, 263-266, Dec 1952. [1]
• 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 [2]
• 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.