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Golden Ratio

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Golden Ratio
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Fig. 1. Vesica Pisces form when unit-one radii circles overlap while tangent to the sides of a double unit-one square. The intersections of the circles are separated by the square root of three. Click for a large image. © 2010 DEM Copy freely with copyright intact, for non-commercial purposes only.


A Greek letter, pronounced –fy–which rhymes with –pi–

(many people say fee, which is OK too)

Lower case Greek letter phi is φ
Upper case Greek leter Phi is Φ

Recent amateur discovery in solid geometry

Orthogonal angles viewed from the center-point, from the torus plane, of a special torus knot
Golden orthogonal torus knots

See: Golden Orthogonal Torus Knots

Circa 2013
DonEMitchell (talk) 09:19, 2 February 2014 (MST)


The Golden Ratio, notated by the Greek letter Phi (Φ) equals one plus the square root of five, all divided by two, which evaluates to 1.618... (continues as an irrational number).

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

—Euclid, Elements  Book VI
The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form. ;—Adolf Zeising 1854

Article: Golden Ratio Discovered in the Quantum World
By Rakefet Tavor, Epoch Times Staff


“The "golden ratio," which is equal to approximately 1.618, can be found in various aspects of our life, including biology, architecture, and the arts. But only recently was it discovered that this special ratio is also reflected in nanoscale, thanks to researchers from the U.K.'s Oxford University, University of Bristol, and Rutherford Appleton Laboratory, and Germany's Helmholtz-Zentrum Berlin for Materials and Energy (HZB).
Their research, published in the journal Science on Jan. 8, examined chains of linked magnetic cobalt niobate (CoNb2O6) atoms only one atom wide to investigate the Heisenberg Uncertainty Principle. They applied a magnetic field at right angles to an aligned spin of the magnetic chains to introduce more quantum uncertainty. Following the changes in field direction, these small magnets started to magnetically resonate. Neutrons were fired at the cobalt niobate atoms to detect the resonant notes.
'We found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618... which is the golden ratio famous from art and architecture,' said principal researcher Dr. Radu Coldea of Oxford University in a press release. 'It reflects a beautiful property of the quantum system —a hidden symmetry.' Dr. Alan Tennant, who led the research group in Berlin, said: 'Such discoveries are leading physicists to speculate that the quantum, atomic scale world may have its own underlying order. Similar surprises may await researchers in other materials in the quantum critical state.'”
The research paper is available at (

Source :

Phi, Pi, and Fibonacci Numbers

This relationship was derived after Oberg noticed an interesting relationship between pi and phi while contemplating geometric questions related to the location of the King and Queen’s burial chambers in the Great Pyramid, Cheops, of Giza, Egypt, the design of which is based on phi.


Golden Ratio as a resistance value on a resistor network

Infinite resistor network 598x215.png
A physical system which demonstrates the principles of continued fractions and rational approximants is the infinite resistor network shown in Fig. 7.1. The total resistance of the network, R, is given by

R=[R1,R2,R3,…] (7.22)

In the case where all resistors are 1 ohm then the resistance is given by  R = [1,1,1,…] and from Eq. (7.10) a resistance of τ ohms is obtained.

The Golden Ratio and Fibonacci Numbers, R. A. Dunlap. 1997, World Scientific, ‎Mathematics

Note: τ is Dunlap's preferred symbol for the Golden Ratio.
The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel
by Sameen Ahmed KHAN
* (PDF) paper: URL:

Phi in 3D

3D Phi Geometry R. Knott
Phi Brick by DonEMitchell with Google Sketchup 8 (free version).

Phi Brick

Each dimension of the rectangle, the lengths of each side, is a power of the golden ratio (Phi):

0.6180339887498948482045868343656... = Phi –1 (an irrational number of unending decimal digits)
1.000 = Phi 0 (Let's check Pythagoras for a power-rule in operation among Phi degrees (powers).
1.6180339887498948482045868343656... = Phi 1 (an irrational number of unending decimal digits)
  • The unit dimensions of the Phi brick illustrated are Phi to the powers of -1, 0, 1 which produces a diagonal of the brick at 2.0 units long, or 2Phi 0.
  • The length of the solid diagonal (Lsolid) is found with the Pythagoras theorem in 3D:

Lsolid = sqrt( Phi(n-1)2 + Phin2 + Phi(n+1)2 )
sqrt(Lsolid) = [Phi-1]2 + [Phi0]2 + [Phi1]2
sqrt(Lsolid) = [Phi-1 × Phi-1] + [Phi0 × Phi0] + [Phi1 × Phi1]
sqrt(Lsolid) = Phi(-2) + Phi(0) + Phi(2)
sqrt(Lsolid) = 0.38196… + 1 + 2.61803… = 4
Lsolid = sqrt( 4 ) = 2

See also: Phi Brick R. Knott

Fibonacci Numbers between the Intergers

Exact Golden Ratio Formula

Mathematically, the irrational value of the Golden Ratio, which is a continuing decimal to notate numerically, can be expressed to any degree of decimal precision if one is able to use sufficient precision of the square-root-of-five, as discovered (likely) by François Édouard Anatole Lucas, circa the late horse-back era (1842–1891).

Lucas Numbers and Fibonacci Numbers are both popular names for different summation sequences. The Fibonacci Sequence begins with the numbers zero (0) and one (1) summed, to produce two (2). By summing the result back into an accumulating sum, the summing-process produces a list of integer numbers.

The Lucas Numbers begin with the numbers two (2) and one (1), while the Fibonacci Sequence begins with zero (0) and one (1).

Any two neighboring-numbers in any summation sequence will divide into each other to produce an approximate value of the Golden Ratio. See the next section for approximations.

A numerically exact value of the Golden Ratio may be produced with the formula attributed to François Lucas.

Φn = (L_n + F_n * √5) / 2
Φ = ( (L_n + F_n * √5)/2 )(1/n))

Sequence Number (n) Lucas Number Fibonacci Number Golden Ratio Formula
n=0 2 0 ((Ln + Fn*√5) / 2)(1/0)
[invalid: div/0]
n=1 1 1 ((1 + 1*√5) / 2)(1/1)
n=2 3 1 ((3 + 1*√5) / 2)(1/2)
n=3 4 2 ((4 + 2*√5) / 2)(1/3)
n=4 7 3 ((7 + 3*√5) / 2)(1/4)
n=5 11 5 ((11 + 5*√5) / 2)(1/5)
n=6 18 8 ((18 + 8*√5) / 2)(1/6)

Golden Ratio Approximations

Formula Value Φ - Formula
(5π/6)1/2 [1] 1.6180215937964160450333550140534… +0.00001239
(7π/5e) [1][2] 1.6180182897072904050743015698393… +0.00001570
1 + (π / 4)2 1.6168502750680849136771556874923… +0.00118371
Fibonacci quotients
5 / 3 1.6666666666666666666666666666667… -0.04863268
8 / 5 1.6000… +0.01803399
13 / 8 1.625000… -0.00696601
21 / 13 1.6153846153846153846153846153846… +0.00264937
34 / 21 1.6190476190476190476190476190476… -0.00101363
55 / 34 1.6176470588235294117647058823529… +0.00038693
89 / 55 1.6181818181818181818181818181818… -0.00014783
144 / 89 1.6179775280898876404494382022472… +0.00005646
233 / 144 1.6180555555555555555555555555556… -0.00002157
377 / 233 1.6180257510729613733905579399142… +0.00000824


Series on Knots and Everything - Vol. 22 by Alexey Stakhov assisted by Scott Olsen (Central Florida Community College, USA)
Dr. R. Knott's website as a PDF (July 31, 2009)
Financial forcasting with Phi degree periods

Golden Fibonacci Goniometry, Fibonacci-Lorentz Transformations, and Hilbert’s Fourth Problem
URL: Retrieved May 5, 2011
(the article was published in «Congressuss Numerantium, Vol. CXCIII, Dec. 2008 )
This study is devoted to the development of the “golden” Fibonacci goniometry and new approach to Lorentz transformations, which are used in Einstein's special theory of relativity. We propose Fibonacci-Lorentz transformations, which are based on the "golden" Fibonacci goniometry and symmetric hyperbolic Fibonacci functions which in turn are based on the golden mean - the world’s oldest scientific paradigm of harmony and beauty. We obtain a cosmological interpretation of change at the velocity light before, at the moment of, and following the bifurcation, known as the Big Bang. The article also presents the authors’ results on the creation of an infinite set of isometric models of Lobachevski’s plane, which are based on the use of the hyperbolic Fibonacci -functions, where is any real number, in particular, symmetric hyperbolic Fibonacci functions with, which is directly relevant to the Hilbert’s Fourth Problem.

See also


  1. 1.0 1.1 Weisstein, Eric W. "Golden Ratio Approximations." From MathWorld--A Wolfram Web Resource.
  2. e = 2.71828 18284 59045 23536