Mathematics is the study of numbers, quantity, space, pattern, 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.
Turing is often considered to be the father of modern computer science. Turing provided an influential formalisation of the concept of the algorithm and computation with the Turing machine, formulating the now widely accepted "Turing" version of the Church–Turing thesis, namely that any practical computing model has either the equivalent or a subset of the capabilities of a Turing machine. With the Turing test, he made a significant and characteristically provocative contribution to the debate regarding artificial intelligence: whether it will ever be possible to say that a machine is conscious and can think. He later worked at the National Physical Laboratory, creating one of the first designs for a stored-program computer, although it was never actually built. In 1947 he moved to the University of Manchester to work, largely on software, on the Manchester Mark I then emerging as one of the world's earliest true computers.
During World War II, Turing worked at Bletchley Park, Britain's codebreaking centre, and was for a time head of Hut 8, the section responsible for German Naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine which could find settings for the Enigma machine.
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 solutions to the Lorenz system, a set of three differential equations used by mathematician and meteorologist Edward N. Lorenz as a simple description of fluid circulation in a shallow layer (of liquid or gas) uniformly heated from below and cooled from above. To be more specific, the figure is set in a three-dimensional coordinate system whose axes measure the rate of convection in the layer (x), the horizontal temperature variation (y), and the vertical temperature variation (z). As these quantities change over time, a path is traced out within the coordinate system reflecting a particular solution to the differential equations. Lorenz's analysis revealed that while all solutions are completely deterministic, some choices of input parameters and initial conditions result in solutions showing complex, non-repeating patterns that are highly dependent on the exact values chosen. 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. One implication is that computing such chaotic solutions to the Lorenz system (i.e., with a computer program) to arbitrary precision is not possible, as any real-world computer will have a limitation on the precision with which it can represent numerical values. The particular solution plotted in this animation is based on the parameter values used by Lorenz (σ = 10, ρ = 28, and β = 8/3, constants reflecting certain physical attributes of the fluid). Note that the animation repeatedly shows one solution plotted over a specific period of time; as previously mentioned, the true solution never exactly retraces itself. Not all solutions are chaotic, however. Some choices of parameter values result in solutions that tend toward equilibrium at a fixed point (as seen, for example, in this image). Initially developed to describe atmospheric convection, the Lorenz equations also arise in simplified models for lasers, electrical generators and motors, and chemical reactions.