Identities of Phi

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Phi is the Greek letter, Φ upper, and φ lower case, used frequently in mathematics to refer to the golden ratio.
Phi equals the quantity five raised to the power of 1/2, multiplied by 1/2, and added to 1/2, or
Φ = 50.5 × 0.5 + 0.5 = 1.6180339887498948482045868343656…
More traditionally, Φ and φ are notationally expressed as:
Φ = (√5 + 1)/2 = 1.6180339887498948482045868343656…
And:
φ = (√5 - 1)/2 = 0.6180339887498948482045868343656…



Only a small sample follows.

See


Connection of Adjacent Degrees of Phi

Φn = Φn-1 + Φn-2

Where
φ = Φ - 1 = 0.618…
n is the ordinal position in the Fibonacci sequence
Source: Mathematics of Harmony[1]


Representation of the n-th Degree of Phi by Fibonacci and Lucas Numbers

Φn = (Ln + Fn√5 ) / 2

Where
φ = Φ - 1 = 0.618…
n is the ordinal position in the Fibonacci sequence
Ln refers to a number of the Lucas sequence (2, 1, 3, 4, 7, 11, 18...) at position n.
Fn refers to a number of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) at position n.
Source: Mathematics of Harmony[2]



  1. Mathematics of Harmony Alexey Stakhov, Pg. 91 (2.56) World Scientific Publishing Co. Pte. Ltd., Singapore 2009
  2. Mathematics of Harmony Pg. 91 (2.60)