A Platonic solid
is a convex regular polyhedron
. These are the three-dimensional analogs of the convex regular polygons
. There are precisely five such figures (shown on the left). The name of each figure is derived from the number of its faces: respectively 4, 6, 8, 12 and 20. They are unique in that the sides, edges and angles are all congruent
Due to their aesthetic beauty and symmetry, the Platonic solids have been a favorite subject of geometers for thousands of years. They are named after the ancient Greek philosopher Plato who claimed the classical elements were constructed from the regular solids.
The Platonic solids have been known since antiquity. The five solids were certainly known to the ancient Greeks and there is evidence that these figures were known long before then. The neolithic people of Scotland constructed stone models of all five solids at least 1000 years before Plato.
The Lorenz attractor is an iconic example of a strange attractor in chaos theory. This three-dimensional fractal structure, resembling a butterfly or figure eight, reflects the long-term behavior of a set of solutions to the Lorenz system, three differential equations used by mathematician and meteorologist Edward N. Lorenz as a simple description of fluid circulation in a shallow layer heated uniformly from below and cooled uniformly from above. Analysis of the system revealed that although the solutions are completely deterministic, they develop in complex, non-repeating patterns that are highly dependent on the exact values of the parameters and initial conditions. As stated by Lorenz in his 1963 paper Deterministic Nonperiodic Flow, "Two states differing by imperceptible amounts may eventually evolve into two considerably different states". He later coined the term "butterfly effect" to describe the phenomenon. The particular solution plotted in this animation is based on the parameter values used by Lorenz (σ = 10, ρ = 28, and β = 8/3). Initially developed to describe atmospheric convection, the Lorenz equations also arise in simplified models for lasers, electrical generators and motors, and chemical reactions.