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The Golden Ratio / The Golden Spiral / The Fibonacci Sequence

StoryAlity Spirality

“The knowledge of which geometry aims is – the knowledge of the eternal.”

Plato

Plato suggested there are ideal and perfect forms, and – sometimes – they emerge in Nature/The World/Reality/The Universe.

Plato and Nature

Some other great thinkers in history who, at some point have also felt likewise, include (among others):

  • Euclid (Euclid wrote about/defined The Golden Ratio in his Elements, circa 300 B.C.)
  • Fibonacci (Fibonacci noticed the mathematical sequence behind The Golden Ratio  in his mathematical treatise, Liber Abaci (1202 AD))
  • Aristotle (who also preached about The Golden Ratio)
  • Einstein
  • Godel
  • Penrose
  • Galileo
  • Newton
  • Hawking

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THE FIBONACCI SEQUENCE

The Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34… (etc.)

Each number in the recursive sequence is the sum of the two preceding numbers.

The numbers of the sequence are also intimately connected with the golden ratio, or

1 : 1.61803(etc. The number is infinite, and non-repeating).

For example – in mathematical terms, the closest rational approximations to The Golden Ratio (1 : 1.6) are: 2/1, 3/2, 5/3, 8/5. The higher in the sequence, the closer two consecutive numbers of the sequence, divided by each other approaches the golden ratio (approximately 1 : 1.618).

The Golden Ratio

The Golden Ratio

1:1.6 is also known as phi – or the Greek φ – or The Golden Ratio (also The Golden Section).

The Golden Section

The Golden Section

The Fibonacci Rectangle

Fibonacci Rectangle

Fibonacci Rectangle

Golden Spiral and Golden Rectangle

Golden Spiral and Golden Rectangle

Golden Spiral and Golden Rectangle

Golden Spiral and Golden Rectangle Combined:

Two Golden Spirals

Two Golden Spirals (mirrored)

Nautilus shells and animal horns also use the Golden Ratio/Fibonacci Sequence.

Nautilus Shell Golden Spiral

Nautilus Shell – The Golden Spiral

The StoryAlity Syntagm:

The StoryAlity Syntagm (Velikovsky 2012)

The StoryAlity Screenplay Syntagm (Velikovsky 2012)

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A BRIEF HISTORY OF THE GOLDEN RATIO:

“Divine Proportion: Phi In Art, Nature, and Science” (Hemenway 2005) – click to see this text on Amazon

According to Priya Hemenway in “Divine Proportion: Phi In Art, Nature, and Science” (2005) the timeline of The Golden Ratio runs thus:

  • Phidias (490–430 BC) – within the Parthenon statues
  • Plato (427–347 BC) – in Timaeus, describes the Platonic solids, which are related to The Golden Ratio.
  • Euclid (c. 325–c. 265 BC), in Elements, gave the first recorded definition of The Golden Ratio
  • Fibonacci (1170–1250) in Liber Abaci
  • Luca Pacioli (1445–1517) – called the golden ratio the “divine proportion” in his Divina Proportione, which was illustrated by Leonardo Da Vinci.
  • Michael Maestlin (1550–1631) – published the first approximation of the (inverse) Golden Ratio as a decimal
  • Johannes Kepler (1571–1630) proved that the golden ratio was the limit of the ratio of consecutive Fibonacci numbers, and describes the golden ratio as a “precious jewel”:
Johannes Kepler

Johannes Kepler (1571-1630): Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel.”

  • (These two `geometric treasures’ are combined in the Kepler triangle.)
  • Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
Sunflower Golden Spiral
Sunflower seeds and The Golden Spiral/Fibonacci Sequence; also in the structure of plant branches
  • Martin Ohm (1792–1872) is believed to be the first to use the term goldener schnitt (golden section) to describe this ratio, in 1835.
  • Édouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
  • Mark Barr (American mathematician) in 1909 suggests the Greek letter phi (φ), the initial letter of Greek sculptor Phidias’s name, as a symbol for the golden ratio.
  • Roger Penrose (1931-) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.

(Hemenway 2005 pp 20-21)

Vitruvian Man – by Leonardo Da Vinci

Vitruvian Man - by da Vinci

Vitruvian Man – by Da Vinci

Da Vinci was obsessed by The Golden Ratio:
The Mona Lisa (La Gioconda) by Da Vinci

The Mona Lisa (La Gioconda) by Da Vinci

Divine Proportions:
Mona Lisa - Golden Section (Da Vinci)

Mona Lisa (Da Vinci) – and The Golden Section 

An analysis of the story structure of the Top 20 ROI (return on investment) Films:
The Top 20 Audience Reach/Budget Films of the Last 70 Years. Data Source: The-Numbers.com. Analysis: JT Velikovsky

The Top 20 Audience Reach/Budget Films of the Last 70 Years. Data Source: The-Numbers.com. Analysis: JT Velikovsky

- also reveals the Golden Ratio:
The StoryAlity Syntagm (Velikovsky 2012)
StoryAlity Syntagm Beat Sheet

StoryAlity Syntagm – Beat Sheet (Velikovsky 2012)

“The knowledge of which geometry aims is – the knowledge of the eternal.”

Plato

 

…Thoughts/Feedback/Comments?

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JT Velikovsky

High ROI Film/Story/Screenplay Analyst

The above is an adapted excerpt from my doctoral thesis: “Understanding And Exploring The Relationship Between: Creativity; Theories Of Narratology; Screenwriting; And Narrative Fiction Feature Film-Making Practices.” It is presented here for the benefit of fellow screenwriting and filmmaking researchers.

JT Velikovsky is a produced feature film screenwriter and million-selling transmedia writer-director-producer. He has been a professional story analyst for major film studios, film funding organizations, and the national writer’s guild. For more see: http://on-writering.blogspot.com/

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REFERENCES

[1] Priya Hemenway, Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. (2005) pp. 20–21

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9 thoughts on “StoryAlity #52 – The Golden Ratio / The Golden Spiral / The Fibonacci Sequence

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