Imploding Phi

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Quotient Impulse of Imploding Phi Degrees


The Fibonacci Sequence is a summation series beginning with a pair of ones. New integers are added to form the sequence by adding the last two numbers of the sequence to produce a new number in the sequence.

A remarkable feature of the Fibonacci Sequence is that the quotient of any adjacent pair in the sequence approximates the value of Phi (Φ = 1.618...). The deeper into the sequence, the closer the approximation is to the value of Φ.

Generalized Application

Interestingly, any two numbers (integers or real) used as the starting pair of numbers in a summation sequence will produce an adjacent pair quotient that also approximates Φ.

An identity of degrees of Phi (powers of Phi) states that a degree of Phi equals the sum of the two lesser degrees[1].

The generalized Fibonacci summation sequence will produce a series of oscillating values of decreasing Phi degrees alternating on either side of zero, if one of the starting pair is of an opposite sign and an adjacent degree of Phi.[2] See Fig. 1.

Fig. 1 Imploding degrees of Phi in a generalized Fibonacci Sequence starting with -Phi14 and Phi13
Fig. 2 An impulse in the adjacent pair quotients of a generalized Fibonacci Sequence beginning with -Phi14 and Phi13.

NOTE: the impulse event of Fig. 2 is actually an approach to positive and negative infinity from either side of 0.500 on the horizontal scale.

Algebraically, an imploding sequence of descending, alternating degrees of Phi (Phi5, -Phi4, Phi3, -Phi2, Phi1, -Phi0, Phi-1, -Phi-2, Phi-3...) would continue forever toward smaller degrees. However, any machine will have a finite precision, and introduce rounding errors. The graphs above were created with 15 decimal digits of precision from the generalized sequence calculated by Google Docs spreadsheet found at

The curious impulse of the pair-quotient graph of Fig. 2 results when decreasing value-quotients loose accuracy as they approach zero, and the remaining digits no longer represent the self-similarity quality of Phi degrees. At this point the quotient-asymptote of -Phi-1 (-0.618...) is departed from, and the normal oscillation approaching an asymptote of +Phi1 (1.618...) begins.

DonEMitchell February 10, 2010.

Argand Plots

Plotting Fibonacci functions on an Argand Diagram
by Dr. R. Knott

See also

The Golden Section - the Number and Its Geometry
by Dr. R. Knott
Map Cycle
Card Colm by Colm Mulcahy
Gibonacci Braclets from modulo patterns in a Generalized Fibonacci (Gibonacci) Sequence, Number magic, and more.
Google search


  1. Identities of Phi —Connection of Adjacent Degrees of Phi
  2. Fibonacci Numbers Extension to Negative Integers (