Platonic+Solids

__Platonic Solids __ by Natalie Gerardi Hexahedron, tetrahedron, octahedron, dodecahedron, and icosahedron; these might sound like the names of bizarrely complex mathematical figures, but in reality, they aren’t. These are the Platonic solids: five shapes that share a few distinguishable features. But, what exactly are the Platonic solids? How are they special? What are they used for? In my research project, these are the questions I tried to answer. To truly understand these unique solids, it is important to look at their history. No one can say exactly when the Platonic solids were first discovered, but we do know the Ancient Greeks studied them extensively. Although some mathematicians credit Pythagoras or Theaetetus for the solids’ “discovery”, they were named for the Greek philosopher/mathematician Plato because of his thorough study of them in his dialogue, Timaeus. Not only did he explain their features, but he also associated each solid with a different element according to their properties: cube (hexahedron) with earth, octahedron with air, icosahedron with water, tetrahedron with fire, and dodecahedron with “heavenly matter”. Contrary to what you might believe, the criteria for a Platonic solid isn’t very complicated. To be a Platonic solid, the object must be a convex solid with faces that are regular convex polygons. Still, only five shapes in the universe fit this criteria. The cube, also known as a hexahedron, has six faces, eight corners, and twelve edges. The tetrahedron has four faces, four corners, and six edges. Another solid, the octahedron, has eight faces, six corners, and twelve edges. The dodecahedron has twelve faces, twenty corners, and thirty edges. The icosahedron completes the group with twenty faces, twelve corners, and thirty edges. Two fascinating ‘tricks’ that are part of the amazing nature of the Platonic solids are stellation and duality. To “stellate” a solid means to extend its faces until they intersect to form a new enclosing shape. This new shape often resembles a star-like object. Duals are the centers of faces of the solids that have common edges. Thus, if two solids are “dual” with each other, you can put one inside the other. Coincidentally, each Platonic solid’s dual is another Platonic solid: hexahedron-octahedron, dodecahedron-icosahedron, and tetrahedron-tetrahedron. Centuries after the Platonic solids began to be studied, you can still see they are used and appear in nature, art, and mathematics. Some crystal structures are in the shapes of Platonic solids, as well as viruses; the herpes virus is a duplicate of an icosahedron. Platonic solid dice have been used since antiquity; some have been found in Etruscan ruins. Many artists, such as M.C. Escher, have used stellated Platonic solids to create beautiful artworks. The solids have also inspired mathematicians to prove their seemingly impossible existence with famous geometric and topological proofs. Thus, Platonic solids are unique, magnificent shapes that will always be examined and marveled at for years to come.

Works Cited Holden, Alan. __Shapes, Space, and Symmetry__. New York: Dover Publications, Inc., 1971. Pappas, Theoni. __The Joy of Mathematics__. San Carlos: Wide World Publishing, 1986. - - -. __The Magic of Mathematics__. San Carlos: Wide World Publishing, 1994.