Polyhedra

Summary: Regular solid shapes play important roles in nature and geometry.

People frequently encounter objects in polyhedral shapes, such as buildings that have cubic or prismatic shapes and geodesic domes or dice that are shaped like polyhedra. This prevalence is partly because of their aesthetic appeal and partly because of their practical properties. Polyhedra also appear in nature; many crystals have the shapes of regular solids, particularly of tetrahedron, cube, and octahedron, and virus capsids can be icosahedral. Furthermore, carbon atoms can form a type of molecule known as “fullerenes,” which are in the form of a triangulated truncated icosahedron. A polyhedron is a solid in space with polygonal faces that are joined along their edges. If the faces consist of regular polygons, then it is called a “regular polyhedron.” A polyhedron is convex if the line segment joining any two points lies on or inside it. Regular convex polyhedra are particularly important for their aesthetic value, symmetry, and simplicity. There are only five of them: the tetrahedron, cube or hexahedron, octahedron, dodecahedron, and icosahedron. Beginning in primary school, students investigate and classify geometric shapes, including polyhedra. In middle school and high school, students explore area and volume measurements as well as transformations and cross-sections.

History

Some of the earliest known polyhedra are the Egyptian pyramids. The five regular solids appear as decorations on Scottish Neolithic carved stone balls, which date to 2000 b.c.e. There are also examples of cuboctahedra worn by east-African women around the ankle and a variety of polyhedral earrings in medieval Europe. The Greeks are thought to have first studied the mathematical properties of regular solids, particularly the Platonic solids, named for Plato. The last book of Euclid of Alexandria’s Elements is devoted to the study of the properties of these solids, including detailed descriptions of their construction. The book is based on the work of Theaetetus of Athens. There is some evidence that Hippasus of Metapontum may have been the first to describe the dodecahedron. Hypsicles of Alexandria inscribed regular polyhedra in a sphere in his treatise. The Platonic solids also represented physical aspects: Earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron, and the dodecahedron with the universe. Plato noted: “So their combinations with themselves and with each other give rise to endless complexities, which anyone who is to give a likely account of reality must survey.”

The Kepler–Poinsot polyhedra are named for the 1619 work of Johannes Kepler and the 1809 work of Louis Poinsot. They constructed four regular “stellated” polyhedra. These new solids were obtained by extending the faces. In the twentieth century, Donald Coxeter classified and studied the stellation process and described many stellated polyhedra.

Properties

One common classroom investigation that relates to polyhedra is the Euler characteristic Χ, named for Leonhard Euler. It is an equation that combines the number of vertices (V), edges (E), and faces (F) of a polyhedron as γ=V-E+F. All convex polyhedra have the same Euler characteristic: 2. René Descartes discovered the polyhedral formula in 1635, and Euler discovered it in 1752. In the nineteenth century, Ludwig Schläfli generalized the formula to polytopes and Henri Poincaré proved the result.

The shape of a polyhedron lends itself to a very convenient symbolic or combinatorial description, called the “Schläfli symbol” of the polyhedron. Let {n, p} represent a regular polygon with n-gon faces, p of them meeting at each vertex. For example {4, 3} would represent a cube because three squares meet at each vertex This symbolic representation is particularly useful if one would like to express various quantities like the dihedral angle, angular deficiency, radii of inscribed and circumscribed spheres, and surface area. For instance, the surface area of a Platonic solid {n, p} can be expressed by

where F is the number of faces and a is the side length.

Mathematically, polyhedra are very appealing for their fine properties such as duality, symmetry, and versatile constructability. The dual of a polyhedron is constructed by taking the vertices of the dual to be the centers of the faces of the original figure by interchanging faces and vertices. For instance, the dodecahedron and the icosahedron are duals. Many polyhedra are highly symmetrical, and in the nineteenth century, Felix Klein investigated them. The groups of symmetries are algebraic structures consisting of reflections and rotations. One can also generate new polyhedra from old by truncating the vertices of polyhedra, a process known and studied since antiquity. Some of the truncated polyhedra are also known as the “Archimedean solids,” named for Archimedes of Alexandria, whose faces consist of two or more types of regular polygons.

There are 13 Archimedean solids, and there are 53 other semiregular, non-convex polyhedra, which are non-Archimedean. The collection of all Platonic, Kepler–Poinsot, Archimedean, and semiregular, non-convex polyhedra together with prisms form the family of polyhedra called “uniform polyhedra.”

Non-Euclidean polyhedra took on a prominent role in some theories of a spherical dodecahedral universe at the beginning of the twenty-first century. There are also non-Euclidean polyhedra with no flat equivalents. For instance, a spherical hosohedron with Schläfli symbol {2, n} is shaped like a segmented orange or beach ball with lune faces. The name “hosohedron” is attributed to Coxeter.

There have been many artistic and physical models of polyhedra in mathematics classrooms. With the advent of perspective, polyhedra were easier to draw and mathematicians and artists designed and collected polyhedral models. Albrecht Dürer introduced polyhedral nets in his 1525 book. Students continue to use nets to build models. In 1966, Magnus Wenninger published a work on polyhedral models for the classroom through the National Council of Teachers of Mathematics. Wenninger noted that the popularity of the book reflected the continued interest in polyhedra. In the twenty-first century, origami polyhedra have also become important in mathematics and computer science classrooms and research.

Bibliography

Artmann, Benno. “Symmetry Through the Ages: Highlights From the History of Regular Polyhedra.” In Eves’ Circles. Edited by Joby Anthony. Washington, DC: Mathematical Association of America, 1994.

Cromwell, Peter. Polyhedra. New York: Cambridge University Press, 1997.

Demaine, Erik, and Joseph O’Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. New York: Cambridge University Press, 2007.

Gabriel, Francois. Beyond the Cube: The Architecture of Space Frames and Polyhedra. Hoboken, NJ: Wiley, 1997.