The Golden Ratio , By Ioannis Fanourakis

1. The golden ratio (symbol is the Greek letter "phi") is a special number
approximately equal to 1,618 It appears many times in nature ,geometry, art,
architecture and other areas. The term "phi" was coined by American
mathematician Mark Barr in the 1900s. The Golden Ratio is also sometimes
called the golden section, golden mean, golden number, divine
proportion, divine section and golden proportion.

2. We find the golden ratio when we divide a line into two parts so that:
the whole length divided by the long part
is also equal to
the long part divided by the short part
The Idea Behind It

3. 𝒂
𝒃
=
𝒂 + 𝒃
𝒂
⇔
𝒂
𝒃
=
𝒂
𝒂
+
𝒃
𝒂
⇔
𝝋 = 𝟏 +
𝟏
𝝋
⇔ 𝝋𝟐
= 𝝋 + 𝟏 ⇔ 𝝋𝟐
− 𝝋 − 𝟏 = 𝟎
𝝋 =
𝟏 − 𝟓
𝟐
𝒐𝒓 𝝋 =
𝟏 + 𝟓
𝟐
But 𝝋 >ο so
𝝋 =
𝟏+ 𝟓
𝟐
= 1,61803398874989484820... (etc.)

6. Golden Triangles

7.  There is a special relationship between the Golden Ratio and
the Fibonacci sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
(The next number is found by adding up the two numbers before it.)
 This sequence ties directly into the Golden ratio because if you take any
two successive Fibonacci numbers, their ratio is very close to the Golden
ratio
 These numbers can be applied to the proportions of a rectangle, called the
Golden rectangle.
 The Golden rectangle is also related to the Golden spiral, which is created
by making adjacent squares of Fibonacci dimensions.

11. The golden ratio in art
Some artists and architects believe the Golden Ratio makes the
most pleasing and beautiful shape.
Many buildings and artworks have the Golden Ratio in them,
such as the Parthenon in Greece, but it is not really known if it
was designed that way.

12. The Taj Mahal
Notre Dame

14. The Golden ratio was used to achieve balance and beauty in many
Renaissance paintings and sculptures.

15. Da Vinci used the Golden ratio to define all of the proportions in his
Last Supper, including the dimensions of the table and the
proportions of the walls and backgrounds.

16. The Golden ratio also appears in da Vinci's Vitruvian Man, the Mona
Lisa and many other art works.

17. Other artists who employed the Golden ratio are :
Michelangelo

18. Seurat

19. Raphael

20. Rembrandt

21. Salvador Dali

22. Seed heads: The seeds of a flower
are often produced at the center and
migrate outward to fill the space. For
example, sunflowers follow this
pattern.
Flower petals: The number of petals
on some flowers follows the
Fibonacci sequence. It is believed that
in the Darwinian processes, each
petal is placed to allow for the best
possible exposure to sunlight and
other factors.

23. Tree branches: The way tree
branches form or split is an example
of the Fibonacci sequence.
Shells: Many shells, including snail
shells and nautilus shells, are perfect
examples of the Golden spiral.

24. Spiral galaxies: The shape of the spiral is identical to
the Golden spiral, and the Golden rectangle can be
drawn over any spiral galaxy.
Hurricanes: Much like shells, hurricanes often display
the Golden spiral.

25. Animal bodies: The measurement of the human navel to the floor and the top
of the head to the navel is the Golden ratio.

26. But we are not the only examples of the Golden ratio in the animal
kingdom;

27. The golden ratio exists in many things that mother nature has given us!
But not only there. Through the years humans have been inspired by
the golden ratio and decided to add it in their daily lives.
Big factories, websites or logo creators use the golden ratio as the main
foundation of their designs in order to look harmonious. Some
examples are:

28. Golden ratio in logo design

29. Apple

30. Pepsi

31. Twitter

32. Mc Donalds

33. Golden Ratio in web design

34. Not only website and logo creators were inspired by golden ratio, but also
during the last years it seems that animators have decided to work on their
sketches with it!
For example:
Kim Possible
Sonic

35. Golden Ratio in photography

39. People since ancient times have tried to connect music to Mathematics and many
believe that there is, indeed, a strong connection. In the 17th century, Gottfried
Leibniz wrote that “ music is the pleasure the human mind experiences from
counting without being aware that it is counting”.
We can commonly observe the usage of the golden ration in many music pieces
and not only there. It is observed on the way musical instruments are made,
music pieces are written, in speaker wires and in the acoustic design of some
cathedrals (just to name a few).
But we don’t know if musicians used it subconsciously or consciously. Often
theorists suggest that artists used the golden ratio, whereas they didn’t, usually
due to miscalculation. Experts claim, however, that Beethoven, Bartók, Debussy,
Schubert, Bach, Satie etc. applied the golden ratio to write their sonatas, but no
one is exactly sure why it worksed so well. Debussy is, actually, known for
using the golden ratio. In a letter of August 1903 he writes to his publisher:

40. Mozart and the Golden Ratio
• Mozart wrote some of the most beautiful piano concertos. Within these
pieces of music, Mozart implemented the Fibonacci sequence.
• Mozart wrote many well known sonatas. A classical sonata is composed of
three parts. The Exposition, the Development, and the Recapitulation. The
exposition is where the main musical theme is introduced and the
Development and Recapitulation is where the theme is developed.

41. Mozart composed his famous sonatas so that the movement from the Exposition
to the Development and Recapitulation was at the Golden Ratio. For example,
the Mozart Sonata 279, No. 1 contained a total of 100 measures. The first
movement was 38 measures for the Exposition. The second and third
movements were 62 measures. The ratio of 62 to 100 is 0.618 which equals
exactly the golden proportion. A Table of Mozart’s piano sonatas was published
in a book by Alfred S. Posamentier and Ingmar Lehman

42. Fibonacci Series was the basis for the proportions that Antonio Stradivari used to
construct his eponymous violins.
It starts with a mathematical pattern of increasing values. They are 0, 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, 144, 233 and so on; each number represents the sum of the two
preceding it. Taking any one number from the series above 5 – let’s say 21 – and
dividing it by the previous number (in this case 13), the ratio is 1.6. This is referred
to as the Golden Ratio.
Stradivarius Violins
Fibonacci Sequence for the construction of the violin

43. There is also a connection between music and the Fibonacci numbers (1, 1, 2, 3,
5, 8, 13, 21…)
An octave on the piano consists of 13 notes. 8 are white keys and 5 are black
keys.
A scale is composed of 8 notes, of which the 3rd and 5th notes create the
foundation of a basic chord.
In a scale, the dominant note is the 5th note, which is also the 8th note of all 13
notes that make up the octave

44. The bass clef looks alike with the golden spiral!

45. The golden ratio is , moreover, spotted in modern music pieces.

46. Drake’s “In my feelings” also has a phi
moment.

51. https://www.mathsisfun.com/numbers/
golden-ratio.html
https://www.goldennumber.net/
https://www.canva.com/learn/what-is-
the-golden-ratio/#golden-ratio-layout
Sources:
https://www.classicfm.com/
https://fibonacci.com/music/
http://jorgevariego.com/?p=696
https://www.youtube.com/watch?v=9mo
zmHgg9Sk