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Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
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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.
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The Lorenz attractor is an iconic example of a strange attractor in chaos theory. This threedimensional fractal structure, resembling a butterfly or figure eight, reflects the longterm 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, nonrepeating 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.
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