Phi ~ the Golden Mean , Fibonacci , Mandelbrot ~ and The Artist



Phidias – the Golden Mean.

Fibonacci ~ mathematician . Fibonacci Sequence : ” the orderly hand of nature .”

Mandelbrot – mathematician and the geometry of fractals .

The Artist – to the artist a gift of infinite complexity  and  beauty .

This is in no way a learned mathematical treatise , nor a treatise on art and architecture . Rather it is an attempt by the writer and artist to understand the history and discovery of fractals – their complexity and their beauty . And the extraordinary gift of the mathematician to the artist . To the mathematician whose learning I may offend , my apologies . And to those who have made an academic study of art and architecture  a similar apology .  For anyone just interested in the beauty of fractals – but find the concept hard to understand  – if I have added to their insight and understanding  , then  I have achieved what I had hoped .

The Golden Mean or Golden  Ratio  . The Golden Ratio has unique mathematical properties and is defined by the ancient Greek letter Phi . It was used by the Greek sculptor and mathematician Phidias to determine the ratios  in his sculptures . In the simplest of mathematical terms it describes a numerical ratio. Mathematically the definition is more complex, but for the purpose of this post, it can be described as a ratio of harmony , of equilibrium .


The ancient Greeks observed its appearance in nature and the harmonious  repetition of patterns  and scale .The ancient Egyptians are said to have used the ratio of the Golden Mean in the building of the Great Pyramid of Giza .The Golden Mean is used throughout history  . In art and in architecture  . In the work of da Vinci and Michaelangeo .   The Golden Mean is found not only in the work of the great artists  and architects , but in the beauty of nature ~ the formation of a cloud, a nautilus shell , a daisy …

Fibonacci . The famed Italian mathematician was  born in Pisa in 1175 .He was widely travelled and as a mathematician made an extensive study of ancient civilizations and their numerical systems. His sequential system of numbers to determine the perfect ratio means the terms Golden Mean and Fibonacci Sequence are often used together .                  ( Mathematicians – please forgive the imprecise definition .  )

The Fibonacci Sequence has been described as ” nature’s numbering system ” . The wonderful whorls and spirals at the centre of a sunflower are the result of constant numerical repetition .




The. 17th  Century physician and naturalist, Sir Thomas Browne describes the Fibonacci Sequence in nature : ” Right lines and circles make out the… plants ; In the parts thereof we find heliacal or spiral roundles… And cannot overlook the orderly hand of nature .. The seeds of many … flowers diffuse themselves circularly… conformable  to the spider’s web .”

” The orderly hand of nature .” Sir Thomas Browne

All of the above suggest order and regularity . But in  the early 20th.  , a Lithuanian mathematician , Benoit Mandelbrot ( b.1924 ) , found in the harmonious repetitions  of patterns and ratios , irregularities ..  Mandelbrot used the term fractal to describe these irregularities – the word is derived from the Latin ” fractus ” , to break . Mandelbrot ,  known as the father of fractal geometry  used the term to    ” describe objects and surfaces  which are irregular at various dimension of scale .” The particular fractal, named after him, was called  the Mandelbrot set .

Fortuitously, Mandelbrot’s work was concurrent with the development of computers and their ” number crunching ” capabilities . What began to emerge were images ofextraordinary beauty . In pragmatic terms Mandelbrot’s work had application to many of the sciences . And for the artist opened an extraordinary vista of creative exploration .

Mandelbrot Set

Fractured Fractal .  At the time of Mandelbrot’s death in 2010, the Guardian wrote the following tribute to him : ” Few have woven such a beautiful braid of art and science as Mandelbrot … They are icons of modern understanding of the universe’s complexity . . The Mandelbrot Set… with its fizzing fringe of chrystal-like microforms blossoming out of a conjunction of black circles … is the outcome of geometrical calculations… Mandelbrot’s startling fractals created a visual manifesto for a non-Euclidean universe… Mandelbrot was a modern Leonardo … he gave us a visual lexicon for our complex world .” ( Mandelbrot : the man who made geometry an art .” Jonathon Jones The Guardian . )

Fractal  Art .

Fractal Art is popularly defined as  a genre belonging to the 20th  Century and the age of computers .   There are fractal images throughout history – but it is only through Mandelbrot that their complex mathematical structure is seen for the first time .

Mathematics : The following images are to show the beauty of mathematics that surrounds us every day . To these images will then be added  images of fractal art – the dimension they add to the artist’s palette .


Guggenheim Museum ~ Phi.

Fractal Art : These images have been designed by me – the first image derives from the fractal , the second derives from the Latin fractus – to break .

Flaming Fractal       Artist AE




Phideas , Fibonacci  and Mandelbrot .  A tribute to Benoit Mandelbrot. 1924 ~2010

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