Polyhedron

Lore & Background

The origin of polyhedra dates back to the ancient era. Ancient Egypt's four-sided Egyptian pyramids are known for the pyramidal structure, with the study of calculating their volume, specifically the volume of a frustum, in the Moscow Mathematical Papyrus. In Ancient Greek, the Monte Loffa dodecahedron was discovered in Italy made of bronze, and Platonic solids were discovered and studied by Ancient Greek mathematicians, with Plato describing their association to the natures for each in his Timaeus, later treated in Euclid's Elements. In the Renaissance, toroidal polyhedra were used for sketching on polyhedral's perspective views, skeletal models, and nets appearance. Leonhard Euler worked on the polyhedral characteristics and the solution for the Seven Bridges of Königsberg's problem, underlying the field of topology. Johannes Kepler discovered two non-convex regular polyhedra, and in 1809 Louis Poinsot discovered two additional regular star polyhedra (the great dodecahedron and great icosahedron), completing the set of four Kepler–Poinsot polyhedra. Many results on polyhedral concepts include Hilbert's third problem, Steinitz's theorem, and stellation of Platonic solids.

Reader's Guide

Polyhedra are central to geometry and topology, with definitions that vary across contexts. Convex polyhedra are well-defined and include familiar examples like cubes and pyramids. There exist many miscellaneous families, such as space-filling polyhedra, flexible polyhedra, ideal polyhedra in hyperbolic space, lattice polyhedra with integer coordinates, orthogonal polyhedra with edges parallel to Cartesian axes, and polyhedral compounds sharing a center. Polyhedra can be generalized into infinitely many faces called apeirohedra, complex polyhedra in Hilbert space, and forms allowing curved faces and edges. The study of polyhedra has influenced mathematics through Euler's work on polyhedral characteristics and topology, and through discoveries like the Kepler–Poinsot star polyhedra. Polyhedra appear in many fields, including biology, nature, and modern computational geometry. Despite no universal agreement on the definition of polyhedra in general, they are typically understood as solids or surfaces described by vertices, edges, and faces, with a three-dimensional interior volume. Definitions range from naive solids bounded by planes to abstract partially ordered sets, with geometric realizations mapping vertices to points.

Did You Know?

• The term 'polyhedron' comes from Greek 'poly-' (many) and '-hedron' (base, seat).
• Ancient Egypt's Moscow Mathematical Papyrus includes study of calculating the volume of a frustum of a pyramid.
• Johannes Kepler discovered two non-convex regular polyhedra, and in 1809 Louis Poinsot discovered two additional regular star polyhedra (the great dodecahedron and great icosahedron), completing the set of four Kepler–Po
• Leonhard Euler worked on polyhedral characteristics and the Seven Bridges of Königsberg problem, underlying topology.