Portal:Mathematics
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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.
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A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another Image credit: User:PAR |
In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is a mathematical constant, usually denoted by the Greek letter φ (phi).
Expressed algebraically, two quantities a and b (assuming ) are therefore in the golden ratio if
It follows from this property that φ satisfies the quadratic equation φ^{2} = φ + 1 and is therefore an algebraic irrational number, given by
which is approximately equal to 1.6180339887.
At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.
Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, divine proportion (Italian: proporzionedivina), divine section (Latin: sectio divina), golden proportion, golden cut, and mean of Phidias.
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A hypotrochoid is a curve traced out by a point "attached" to a smaller circle rolling around inside a fixed larger circle. In this example, the hypotrochoid is the red curve that is traced out by the red point 5 units from the center of the black circle of radius 3 as it rolls around inside the blue circle of radius 5. A special case is a hypotrochoid with the inner circle exactly one-half the radius of the outer circle, resulting in an ellipse (see an animation showing this). Mathematical analysis of closely-related curves called hypocycloids lead to special Lie groups. Both hypotrochoids and epitrochoids (where the moving circle rolls around on the outside of the fixed circle) can be created using the Spirograph drawing toy. These curves have applications in the "real world" in epicyclic and hypocycloidal gearing, which were used in World War II in the construction of portable radar gear and may be used today in 3D printing.
Did you know -
- ... that, according to the pizza theorem, a circular pizza that is sliced off-center into eight equal-angled wedges can still be divided equally between two people?
- ... that the Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle?
- ... that the Life without Death cellular automaton, a mathematical model of pattern formation, is a variant of Conway's Game of Life in which cells, once brought to life, never die?
- ... that one can list every positive rational number without repetition by breadth-first traversal of the Calkin–Wilf tree?
- ... that the Hadwiger conjecture implies that the external surface of any three-dimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?
- ... that an equitable coloring of a graph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than a graph coloring without this constraint?
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