M.C. Escher (1898-1972) produced work that remains among the most widely reproduced and popular graphic art of the twentieth century. His brain-teasing prints use interlocking shapes, transforming creatures, and impossible architectures to...

A magic cube is a three-dimensional (or higher dimensional) analogue of a magic square. Specifically, an order n magic cube is an n x n x n array of numbers chosen from the set {1,2,...,n3} so that each number appears exactly once and the ...

The Mandelbrot set is the most famous object in the branch of mathematics known as fractals and chaos theory. It was discovered in 1980 by the mathematician Benoit Mandelbrot, who pioneered this relatively new field. A picture of the Mande...

in cartography, systematic representation on a flat surface of features of a curved surface, as that of the Earth. Such a representation presents an obvious problem but one that did not disturb ancient or medieval cartographers. Only when ...

Marin Mersenne was a French monk, mathematician, scientist, and philosopher known for his development of the Mersenne primes, which have played a significant role in number theory for several centuries. Born into a family of laborers in Oi...

Dame Mary Lucy Cartwright's research and contributions to the field of mathematics have spanned more than seven decades. England claims her as one of its most brilliant citizens and has honored her for the past 50 years. As late as 1996, s...

Field of mathematics that incorporates the methods of algebra and calculus—specifically of limits, continuity, and infinite series—to analyze classes of functions and equations having general properties (e.g., differentiability...

In mathematics, induction is a technique for proving certain types of mathematical statements. The induction principle can be illustrated by arranging a series of dominoes in a line. Suppose two facts are known about this line of dominoes....

The makers of mathematics and their modes of practice were transformed during the nineteenth century. New mathematical ideas were publicized in mathematical journals. Professional mathematicians began to meet together to share their resea...

A mathematical model is an equation or system of equations and/or inequalities used to describe data from real world situations and to make predictions from that data. The model may be as simple as a linear function or as complicated as a ...

In logic and mathematics, an argument that establishes a proposition's validity. Formally, it is a finite sequence of formulas generated according to accepted rules. Each formula either is an axiom or is derived from a previously establish...

A matrix, singular for matrices, is a rectangular array of numbers. Matrices naturally arise in describing a special class of functions called linear transformations. But the concept of matrices originated in the work of the two mathematic...

The concept of real space is easy to understand. A point is a common example of one-dimensional real space and a line, an example of two-dimensional real space. The real world in which we live--where objects have length, width, and height-...

A Möbius strip is constructed by twisting one end of a rectangular strip of paper by 180 degrees (a half twist) and joining it to the other end. The result is an example of a one-sided surface. If you draw a line along the surface sta...

Association of numbers with physical quantities and natural phenomena by comparing an unknown quantity with a known quantity of the same kind. Weights and measures are standard quantities with which such comparisons are made. The earliest ...

The median of a data set is the value above which half of the data lie and below which half the data lie. It is considered to be the "middle" number of the data set. If the data set is composed of an odd number of entries, the median is th...

Mersenne numbers are numbers of the form 2p-1 where p is a prime number. If the Mersenne number 2p-1 itself is prime then it is called a Mersenne prime. The first few are 22-1= 3, 23-1 = 7, 25-1= 31, 27-1 = 127 are all prime numbers but th...

Integration is one of the basic operations of calculus. However, its results are not always straightforward; there are some integrals whose solution is not apparent by inspection. Because this is the case, and because closed form and numer...

Michael Francis Atiyah has had a remarkably long and productive career that is based in topology but encompasses such diverse fields as algebraic geometry, differential equations, and theoretical physics. In recognition of three major theo...

Michel Rolle was a largely self-educated mathematician who taught himself both algebra and Diophantine analysis, a method for solving equations with no unique solution. He won early repute when he solved a problem set by the mathematician ...

Émile Borel was one of the most powerful mathematicians of the twentieth century. He displayed great virtuosity in working in a number of different areas of mathematics, particularly complex numbers and functions, but in addition se...

Minkowski space-time is a single, four-dimensional continuum composed of the three coordinates defining position and a fourth dimension of time. It is a concept developed by German mathematical physicist Hermann Minkowski in 1907, and it p...

The Mode of a data set is the entry or entries that occur most frequently in the set. If a data set has two different values occurring most frequently, it is said to be bimodal. For the data set 1, 2, 4, 8, 8, 10, 10, 10, 12, 14, 15, 15, t...

Modular arithmetic is a generalization of odd and even. We say that two integers, x and y, are congruent modulo a third integer n if and only if x - y is divisible by n. For example, 5 and 8 are congruent modulo 3 because 5 - 8 = -3 is div...

(Latin: “method of affirming” and “method of denying”) In logic, two types of inference that can be drawn using a hypothetical proposition—i.e., from a proposition of the form “If p, then q” (symbo...