




A solid with equivalent faces composed of congruent regular convex Polygons. There are exactly five such solids: the Cube, Dodecahedron, Icosahedron, Octahedron, and Tetrahedron, as was proved by Euclid in the last proposition of the Elements.
The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC . In this work, Plato equated the Tetrahedron with the ``element'' fire, the Cube with earth, the Icosahedron with water, the Octahedron with air, and the Dodecahedron with the stuff of which the constellations and heavens were made (Cromwell 1997).
The Platonic solids are sometimes also known as the Regular Polyhedra of Cosmic Figures (Cromwell 1997), although the former term is sometimes used to refer collectively to both the Platonic solids and KeplerPoinsot Solids (Coxeter 1973).
If is a Polyhedron with congruent (convex) regular polygonal faces, then Cromwell (1997, pp. 7778) shows that the following statements are equivalent.
Let (sometimes denoted ) be the number of Vertices, (or ) the number of Edges, and (or ) the number of Faces. The following table gives the Schläfli Symbol, Wythoff Symbol, and C&R symbol, the number of vertices , edges , and faces , and the Point Groups for the Platonic solids (Wenninger 1989).
Solid  Schläfli Symbol  Wythoff Symbol  C&R Symbol  Group  
Cube  3 2 4  8  12  6  
Dodecahedron  3 2 5  20  30  12  
Icosahedron  5 2 3  12  30  20  
Octahedron  4 2 3  6  12  8  
Tetrahedron  3 2 3  4  6  4 
Let be the Inradius, the Midradius, and the Circumradius. The following two tables give the analytic and numerical values of these distances for Platonic solids with unit side length.
Solid  
Cube  0.5  0.70711  0.86603 
Dodecahedron  1.11352  1.30902  1.40126 
Icosahedron  0.75576  0.80902  0.95106 
Octahedron  0.40825  0.5  0.70711 
Tetrahedron  0.20412  0.35355  0.61237 
Finally, let be the Area of a single Face, be the Volume of the solid, the Edges be of unit length on a side, and be the Dihedral Angle. The following table summarizes these quantities for the Platonic solids.
Solid  
Cube  1  1  
Dodecahedron  
Icosahedron  
Octahedron  
Tetrahedron 
The number of Edges meeting at a Vertex is . The
Schläfli Symbol can be used to specify a Platonic solid. For the solid whose faces are
gons (denoted ), with touching at each Vertex, the symbol is .
Given and , the number of Vertices,
Edges, and faces are given by
Minimal Surfaces for Platonic solid frames are illustrated in Isenberg (1992, pp. 8283).
See also Archimedean Solid, Catalan Solid, Johnson Solid, KeplerPoinsot Solid, Quasiregular Polyhedron, Uniform Polyhedron
References
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Ball, W. W. R. and Coxeter, H. S. M. ``Polyhedra.'' Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 131136, 1987.
Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2: Geometry. Cambridge, MA: MIT Press, p. 272, 1974.
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 128129, 1987.
Bogomolny, A. ``Regular Polyhedra.'' http://www.cuttheknot.com/do_you_know/polyhedra.html.
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Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970.
Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 5157, 6670, and 7778, 1997.
Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 7881, 1990.
Gardner, M. ``The Five Platonic Solids.'' Ch. 1 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 1323, 1961.
Heath, T. A History of Greek Mathematics, Vol. 1: From Thales to Euclid. New York: Dover, p. 162, 1981.
Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, 1992.
Kepler, J. Opera Omnia, Vol. 5. Frankfort, p. 121, 1864.
Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 129131, 1990.
Pappas, T. ``The Five Platonic Solids.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 39 and 110111, 1989.
Rawles, B. A. ``Platonic and Archimedean SolidsFaces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios.'' http://www.intent.com/sg/polyhedra.html.
Steinhaus, H. ``Platonic Solids, Crystals, Bees' Heads, and Soap.'' Ch. 8 in Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, 1960.
Waterhouse, W. ``The Discovery of the Regular Solids.'' Arch. Hist. Exact Sci. 9, 212221, 19721973.
Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, 1971.
© 19969 Eric W. Weisstein