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For every fraction p / q (in its lowest terms) there is a Ford circle C[ p / q ], which is the circle with radius 1/(2q 2) and centre at ( p / q , 1 / 2q 2 ). Two Ford circles for different fractions are either disjoint or they are tangent to one another—two Ford circles never intersect.
In linguistics, ordinal numerals or ordinal number words are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on (e.g., "third", "tertiary"). They differ from cardinal numerals, which represent quantity (e.g., "three") and other types of numerals.
A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . A continued fraction is a mathematical expression that can be writen as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple ...
In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643.
In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning (frequency ratio 3 : 2 ); the result is 3 / 2 × [ 80 / 81 ] 1 / 4 = 4 √ 5 ≈ 1.49535, or a fifth of 696.578 cents. (The 12th power of that value is 125, whereas 7 octaves ...
The volume of a sphere with radius r is 4 / 3 πr 3. The surface area of a sphere with radius r is 4πr 2. Some of the formulae above are special cases of the volume of the n-dimensional ball and the surface area of its boundary, the (n−1)-dimensional sphere, given below. Apart from circles, there are other curves of constant width.
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