PHI, the golden ratio

(Also known as the golden mean)

Written by Paul Bourke
May 1990, Updated January 1995

Definition

Break a line segment into two such that the ratio of the whole to the longest segment is the same as the ratio of the two segments. From the diagram below.

The condition can expressed as a/b = 1/a. This can be rearranged and expressed as a quadratic.

There are two solutions, phi-1 and -phi where

This is the original Greek definition, often phi-1 is used instead.

Solution of a quadratic

Normally the quadratic for which phi is the quoted solution is

The solutions being phi and phi-1

Relationships

phi^x+1 = phi^x + phi^x-1

Continued fractions

phi =

phi = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + .....))))

Relationship to the Fibonnaci series

Consider the first order Fibonnaci series

x₀, x₁, x₂ ..... x_i ..... where x_i = x_i-1 + x_i-2 The ratio

This tends to phi as i tends to infinity. That is, the ratio of consecutive terms in such a series approaches phi, this is true independent of the starting points of the series. The zero order series starts with 1 and 1 as below.

1 1 2 3 5 8 13 21 34 55 89 etc

the ratio of consecutive pairs are

1 0.5 0.67 0.6 0.625 0.6154 0.619 0.6176 0.6182 etc

The ratio of terms of this series as converged to 3 decimal places after only 10 terms.

2 dimensional golden ratio

An alternative definition which is the 2D version of the original definition based on the line segment is: "find a rectangle such that when a square is removed the remaining rectangle has the same proportions as the original". The solution to this is a rectangle with the ratio of its sides being phi.

These rectangles can be inscribed in a so called logarithmic spiral also known as equiangular spirals. Such spirals and occur frequently in nature, for example: shells, sunflowers, and pine cones. The limit point of the spiral is called the "eye of God".

Phi Pyramid

Unique sequence

Find an additive series such that

The only solution is the series

Note: the terms of the series describe a 1,phi Fibonnaci sequence.

Phi to 1000 decimal places

1.618033988749894848204586834365638117720309179805762862135448
  622705260462818902449707207204189391137484754088075386891752
  126633862223536931793180060766726354433389086595939582905638
  322661319928290267880675208766892501711696207032221043216269
  548626296313614438149758701220340805887954454749246185695364
  864449241044320771344947049565846788509874339442212544877066
  478091588460749988712400765217057517978834166256249407589069
  704000281210427621771117778053153171410117046665991466979873
  176135600670874807101317952368942752194843530567830022878569
  978297783478458782289110976250030269615617002504643382437764
  861028383126833037242926752631165339247316711121158818638513
  316203840052221657912866752946549068113171599343235973494985
  090409476213222981017261070596116456299098162905552085247903
  524060201727997471753427775927786256194320827505131218156285
  512224809394712341451702237358057727861600868838295230459264
  787801788992199027077690389532196819861514378031499741106926
  088674296226757560523172777520353613936