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  2. Golden ratio - Wikipedia

    en.wikipedia.org/wiki/Golden_ratio

    Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: [42] + = + =. In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates ⁠ φ {\displaystyle \varphi } ⁠ .

  3. Fibonacci sequence - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_sequence

    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. Fibonacci numbers are also closely related to Lucas numbers, which obey the same ...

  4. Golden spiral - Wikipedia

    en.wikipedia.org/wiki/Golden_spiral

    A Fibonacci spiral approximates the golden spiral using quarter-circle arcs inscribed in squares derived from the Fibonacci sequence. A golden spiral with initial radius 1 is the locus of points of polar coordinates (,) satisfying = /, where is the golden ratio.

  5. Generalizations of Fibonacci numbers - Wikipedia

    en.wikipedia.org/wiki/Generalizations_of...

    The ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero and the sequences where the ratio of the two first terms is (). The sequence can be written in the form

  6. Golden triangle (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Golden_triangle_(mathematics)

    A golden triangle. The ratio a/b is the golden ratio φ. The vertex angle is =.Base angles are 72° each. Golden gnomon, having side lengths 1, 1, and .. A golden triangle, also called a sublime triangle, [1] is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:

  7. Lucas number - Wikipedia

    en.wikipedia.org/wiki/Lucas_number

    All Fibonacci-like integer sequences appear in shifted form as a row of the Wythoff array; the Fibonacci sequence itself is the first row and the Lucas sequence is the second row. Also like all Fibonacci-like integer sequences, the ratio between two consecutive Lucas numbers converges to the golden ratio.

  8. Reciprocal Fibonacci constant - Wikipedia

    en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant

    The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers: = = = + + + + + + + +. Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.

  9. Solving the Riddle of Phyllotaxis - Wikipedia

    en.wikipedia.org/wiki/Solving_the_Riddle_of_Phyl...

    Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur in Plants is a book on the mathematics of plant structure, and in particular on phyllotaxis, the arrangement of leaves on plant stems. It was written by Irving Adler, and published in 2012 by World Scientific.