Polyhedron/Polyhedra
Polyhedra are geometric solids in three-dimensional space made up of polygonal faces connected along their edges. They play a significant role across various fields, including mathematics, architecture, biology, chemistry, medicine, physics, and astronomy. Among the most notable polyhedra are the five regular forms known as Platonic solids: the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, each of which has identical faces. Polyhedra can also vary in face types, such as the snub dodecahedron, part of the Archimedean solids named after the mathematician Archimedes.
The study of polyhedra has deep historical roots, with significant contributions from figures like Euclid, Plato, and modern mathematicians who have explored their geometric properties, classifications, and applications. Polyhedra are not just theoretical constructs; they are found in nature, as seen in salt crystals and the structure of certain viruses. The geometric forms also inspire architectural designs, exemplified by R. Buckminster Fuller’s geodesic domes, which highlight the intersection of aesthetic and functional design. In computer graphics and modeling, polyhedra serve as fundamental elements in visual representation. Overall, the exploration of polyhedra continues to be a dynamic area of interest across multiple disciplines.
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Polyhedron/Polyhedra
Polyhedra are important in many fields inside and outside of mathematics, including architecture, biology, chemistry, medicine, physics and astronomy. Polyhedra are geometric solids in space that consist of polygonal faces joined together along their edges. There are five regular polyhedra—the cube, dodecahedron, icosahedron, octahedron and tetrahedron. Each has identical faces.
The dodecahedron has five-sided faces while the others have three-sided faces. In the naming of a specific polyhedra, such as the icosahedron, "hedron" refers to the faces, while "poly," meaning many, has been replaced by a term that quantitatively represents the number of faces, in this case "icosa" for twenty. Please see Figure 1. The regular polyhedra are called the Platonic solids, for Greek mathematician and philosopher Plato.
Other polyhedra have two or more kinds of faces, like the snub dodecahedron (see Figure 2), which is one of the thirteen Archimedean solids. Archimedean solids were named for Greek mathematician Archimedes of Syracuse.
There are many other families of polyhedra, such as pyramids and the Johnson solids, named for American mathematician Norman Johnson. In mathematics and in applications, investigations related to their classification, volume, surface area, dissection, folding, unfolding, truncation, duality, stellation, faceting, packing, and symmetry groups are of interest.
Overview
For millennia, artists and mathematicians have explored the aesthetic and geometric qualities of polyhedra. Mathematical investigations date back to at least ancient Greece. There is disagreement over who first discovered each one of the Platonic solids, although Theaetetus of Athens is noted as the first to geometrically construct all five regular polyhedra to prove that there were no others. The last book of Euclid’s Elements, named for Euclid of Alexandria, is focused on polyhedra. Plato equated the regular polyhedra with the elements and the structure of the universe, and it was Pappus of Alexandria who attributed the Archimedean polyhedra to Archimedes. A 1525 book by German artist Albrecht Dürer contains the earliest known examples of polyhedral nets. French philosopher Rene Descartes, Swiss mathematician Leonhard Euler and many others have classified and investigated polyhedra using methods like the Euler characteristic, a topological invariant that describes a relationship between the vertices, edges and faces.
Polyhedra are important in many fields because naturally occurring objects, such as crystals and viruses, are either shaped like polyhedra or have the same symmetries. For example, salt crystals have a cube-like structure and the human papilloma virus has icosahedral symmetry. Viruses were once thought to have spherical symmetry, but high resolution x-rays and electron microscopy studies showed the icosahedral nature. In 1962 American biologist Donald Caspar and British chemist Aaron Klug introduced the notion of quasi-equivalence to explain the icosahedral structure.
Polyhedra are also found in architecture and design. One notable person in this context is American engineer and architect R. Buckminster Fuller, who popularized geodesic domes. When an experiment on carbon molecules revealed a truncated icosahedral structure, Buckminsterfullerene, the buckyball, was named for him. In physics and astronomy, polyhedra have been used to model physical phenomena and the shape of the universe. Polyhedra are ubiquitous in computer graphics and imaging.
Bibliography
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Atiyah, Michael F., and Paul M. Sutcliffe. "Polyhedra in Physics, Chemistry and Geometry." Milan Journal of Mathematics 71.1 (2003): 33-58.
Comez, Dogan, Sarah J. Greenwald, and Jill E. Thomley. "Polyhedra." Encyclopedia of Mathematics and Society. Eds. Sarah J. Greenwald and Jill E. Thomley. Pasadena, CA: Salem Press, 2011.
Gabriel, Francois. Beyond the Cube: The Architecture of Space Frames and Polyhedra. Hoboken, NJ: Wiley, 1997.
Hilton, Peter, and Jean Pedersen. A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics. New York: Cambridge UP, 2010.
Lauritzen, Niels. Undergraduate Convexity: From Fourier and Motzkin to Kuhn and Tucker. Hackensack, NJ: World Scientific, 2013.
O’Rourke, Joseph. How to Fold It: The Mathematics of Linkages, Origami, and Polyhedra. New York: Cambridge UP, 2011.
Senechal, Marjorie. Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination. New York: Springer, 2013.