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A semiregular Polyhedron whose faces are all Equilateral Triangles. There are an infinite number of deltahedra, but only eight convex ones (Freudenthal and van der Waerden 1947). They have 4, 6, 8, 10, 12, 14, 16, and 20 faces. These are summarized in the table below, and illustrated in the following figures.

$n$ Name
4 Tetrahedron
6 Triangular Dipyramid
8 Octahedron
10 Pentagonal Dipyramid
12 Snub Disphenoid
14 Triaugmented Triangular Prism
16 Gyroelongated Square Dipyramid
20 Icosahedron

The Stella Octangula is a concave deltahedron with 24 sides:

Another with 60 faces is a ``caved in'' Dodecahedron which is Icosahedron Stellation $I_{20}$.

Cundy (1952) identifies 17 concave deltahedra with two kinds of Vertices.

See also Gyroelongated Square Dipyramid, Icosahedron, Octahedron, Pentagonal Dipyramid, Snub Disphenoid Tetrahedron, Triangular Dipyramid, Triaugmented Triangular Prism


Cundy, H. M. ``Deltahedra.'' Math. Gaz. 36, 263-266, 1952.

Freudenthal, H. and van der Waerden, B. L. ``On an Assertion of Euclid.'' Simon Stevin 25, 115-121, 1947.

Gardner, M. Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 40, 53, and 58-60, 1992.

Pugh, A. Polyhedra: A Visual Approach. Berkeley, CA: University of California Press, pp. 35-36, 1976.

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