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This is an accepted version of this page This is the latest accepted revision, reviewed on 13 June 2024. Position of the year within the 19-year Metonic cycle Not to be confused with Golden ratio. Month of January from Calendarium Parisiense (fourth quarter of the 14th c.). The golden numbers, in the leftmost column, indicate the date of the new moon for each year in the 19-year cycle A golden ...
The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer.
Golden number may mean: Golden number (time), a number assigned to a calendar year denoting its place in a Metonic cycle; Golden ratio, an irrational mathematical constant with special properties in arts and mathematics Fibonacci number, a sequence of numbers which converges on the golden ratio
Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
Golden triangles inscribed in a logarithmic spiral. The golden triangle is used to form some points of a logarithmic spiral. By bisecting one of the base angles, a new point is created that in turn, makes another golden triangle. [4] The bisection process can be continued indefinitely, creating an infinite number of golden triangles.
Many works of art are claimed to have been designed using the golden ratio. However, many of these claims are disputed, or refuted by measurement. [1] The golden ratio, an irrational number, is approximately 1.618; it is often denoted by the Greek letter φ .
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[13] [14] These three ways of combining two numbers were all studied in ancient Greek mathematics, and are called the Pythagorean means. [15] Conversely, this can be taken as an alternative definition of the Kepler triangle: it is a right triangle whose edge lengths are the three Pythagorean means of some two numbers.