# Portal:Mathematics

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## The Mathematics Portal

**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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Dodecahedron Image credit: |

**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.

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This spiral diagram represents all **ordinal numbers** less than *ω*^{ω}. The first (outermost) turn of the spiral represents the finite ordinal numbers, which are the regular counting numbers starting with zero. As the spiral completes its first turn (at the top of the diagram), the ordinal numbers approach infinity, or more precisely ω, the first transfinite ordinal number (identified with the set of all counting numbers, a "countably infinite" set, the cardinality of which corresponds to the first transfinite cardinal number, called ℵ_{0}). The ordinal numbers continue from this point in the second turn of the spiral with *ω* + 1, *ω* + 2, and so forth. (A special ordinal arithmetic is defined to give meaning to these expressions, since the + symbol here does not represent the addition of two real numbers.) Halfway through the second turn of the spiral (at the bottom) the numbers approach *ω* + *ω*, or *ω* · 2. The ordinal numbers continue with *ω* · 2 + 1 through *ω* · 2 + *ω* = *ω* · 3 (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through *ω* · 4, and so forth, up to *ω* · *ω* = *ω*^{2} at the top. (As with addition, the multiplication and exponentiation operations have definitions that work with transfinite numbers.) The ordinals continue in the third turn of the spiral with *ω*^{2} + 1 through *ω*^{2} + *ω*, then through *ω*^{2} + *ω*^{2} = *ω*^{2} · 2, up to *ω*^{2} · *ω* = *ω*^{3} at the top of the third turn. Continuing in this way, the ordinals increase by one power of ω for each turn of the spiral, approaching *ω*^{ω} in the middle of the diagram, as the spiral makes a countably infinite number of turns. This process can actually continue (not shown in this diagram) through and , and so on, approaching the first epsilon number, *ε*_{0}. Each of these ordinals is still countable, and therefore equal in cardinality to ω. After uncountably many of these transfinite ordinals, the first uncountable ordinal is reached, corresponding to only the second infinite cardinal . The identification of this larger cardinality with the cardinality of the set of real numbers can neither be proved nor disproved within the standard version of axiomatic set theory called Zermelo–Fraenkel set theory, whether or not one also assumes the axiom of choice.

## Did you know…

- ...that some functions can be written as an infinite sum of trigonometric polynomials and that this sum is called the Fourier series of that function?
- ...that as of April 2010 only 35 even numbers have been found that are not the sum of two primes which are each in a Twin Primes pair? ref
- ...the Piphilology record (memorizing digits of Pi) is 70000 as of Mar 2015?
- ...with a Perrin number denoted P(i), i=1,2,3..., when i is prime then P(i) is composite, being divisible by i?
- ...that Auction theory was successfully used in 1994 to sell FCC airwave spectrum, in a financial application of game theory?
- ...properties of Pascal's triangle have application in many fields of mathematics including combinatorics, algebra, calculus and geometry?

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The **Mathematics WikiProject** is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's **talk page**.

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