The document explores the concept of the golden ratio, traced back to ancient philosophers like Plato and Euclid, defining it mathematically and illustrating its occurrence in nature, art, and architecture. It introduces Fibonacci's contributions to mathematics, particularly through the Fibonacci sequence, and discusses how these principles relate to various forms of design and natural phenomena. Examples include the shapes of plants, human proportions, and works by famous artists, highlighting the pervasive influence of the golden ratio across different fields.

1. Ruhulla Abdul
IFF 1-2 students
Moscow 2015
The Golden Ratio
Financial University under the Government of the Russian
Federation

2. SO WHAT IS A RATIO?
Remember this?

3. 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.
What?
That statement makes no sense to my right brain at all!
OK, I get it about ratio and proportion. But
what’s the “golden” part?
That is a quote from Plato living between 428 and 348 BC in Greece. Euclid had a great
influence on philosophy, mathematics and geometry, who lived in Egypt. Both of these
men were so brilliant and they set the groundwork for millennia of thinkers and
discoverers, as well as later geniuses in the Renaissance and into our modern times.

4. the ratio of the length of the entire line (A)
to the length of larger line segment (B)
is the same as
the ratio of the length of the larger line segment (B)
to the length of the smaller line segment (C).
This happens only at the point where:
A is 1.618 … times B and B is 1.618 … times C.
Alternatively, C is 0.618… of B and B is 0.618… of A.
Just as p is the ratio of the circumference of a circle to its diameter, Φ (phi) is
simply the ratio of the line segments that result when a line is divided in
one very special and unique way. Divide a line so that:

5. So, to continue, here is some algebra for the math heads in our midst.
This is the solution to Plato and Euclid’s line divided into unequal lengths

6. Phi 1.6180339887…..the Golden Number!

7. Fibonacci was born in 1170 in Pisa, and was known as
Leonardo Bonacci or Leonard of Pisa, which was shorted to
Fibonacci, or son of Bonacci. He was a mathematical genius
who wrote, in the early 13th century the Liber Abaci, or Book of
Calculation. Having traveled extensively he concluded that
trying to do arithmetic with Roman Numerals was much more
difficult than using the Hindu-Arabic number system. He spread
the word through his teachings and book. Centuries later this
numerical sequence was named after him. We are lucky to have
been saved by Fibonacci from having to add and subtract, or
horrors, divide using Roman Numerals!
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377....etc.
FIBONACCI

8. Liber Abaci also posed, and solved, a
problem involving the growth of a
population of rabbits based on idealized
assumptions. The solution, generation by
generation, was a sequence of numbers
later known as Fibonacci numbers.
Although Fibonacci's Liber Abaci contains
the earliest known description of the
sequence outside of India, the sequence
had been noted by Indian mathematicians
as early as the sixth century.
THE FIBONACCI RABBITS
This famous problem was first presented to the world in 1202 where we find on pages 123-128 of the
manuscript of Fibonacci’s Liber Abacci

9. Suppose a newly-born pair of rabbits, one male, one female, are put in a
field. Rabbits are able to mate at the age of one month so that at the end
of its second month a female can produce another pair of rabbits.
Suppose that our rabbits never die and that the female always produces
one new pair (one male, one female)every month from the second
month on. The puzzle that Fibonacci posed was...
How many pairs will there be in one year?
The original problem

10. How to make a Golden Rectangle without math

15. We can do this exercise an infinite number of times but I don’t have a
microscope! The divisions go on and on forever, but for our purposes we
are going to stop now….it is sort of tiresome making these golden
rectangles!

16. PUTTING THEM ALL BACK TOGETHER

17. HERE IS A GOLDEN SPIRAL

18. EACH LINE IS ALSO A
GOLDEN RATIO
HERE IS THE INFINITY POINT

19. Suppose you have some stretcher bars for a 16 x 20
canvas, but you want to make Golden Rectangles
instead. How do I find out how long I need to make
the new stretcher bars?
Multiply by 1.618 (yeah, you have to use a
calculator to do this!) So I decide to have one side
be 16 cm. So the other side has to be 25.8888…, or
roughly 26 cm.
Now you have your Golden Ratio 16 to 26
But there is an easier way to make a golden rectangle!

20. Golden Rectangle Calculator
http://www.miniwebtool.com/golden-rectangle-calculator/?a=30

21. Ф in pentagram

22. And for the fanatic Golden Ratio addict
you can buy an instrument to measure
your distances.
Evidently Dentists and Surgeons use these tools to measure body parts and teeth.
Or do it by your own

23. All sorts of interesting things follow this sequence; in
nature, certain sea shells follow the Fibonacci
sequence, as do breeding pairs of rabbits, certain
plants grow leaves in a perfect Fibonacci sequence
and the sunflower seeds follow the Fibonacci spiral
sequence and astronomers see the Fibonacci spiral in
black holes and galaxies! There are hundreds, or
perhaps thousands of examples. Here are a few…

24. The Golden Spiral constructed from a Golden Rectangle

25. http://jwilson.coe.uga.edu/emat6680/parveen/Fib_nature.htm
By dividing a circle into Golden proportions,
where the ratio of the arc length are equal to
the Golden Ratio, we find the angle of the arcs
to be 137.5 degrees. In fact, this is the angle at
which adjacent leaves are positioned around the
stem. This phenomenon is observed in many
types of plants.
PETALARRANGEMENTS

26. In the pinecone pictured, eight spirals can be seen to be
ascending up the cone in a clockwise direction
while thirteen spirals ascend more steeply in a
counterclockwise direction.http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
PINECONES

27. One set of 5
parallel spirals
ascends at a
shallow angle to
the right
a second set of
8 parallel
spirals ascends
more steeply to
the left
and the third set of
13 parallel spirals
ascends very steeply
to the right
http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
PINEAPPLES

28. DAISY
2. we can see the phenomenon in almost two-dimensional form.
3. The eye sees twenty-one counterclockwise 4. and thirty-four logarithmic or equiangular spirals. In any daisy, the
combination of counterclockwise and clockwise spirals generally consists of
successive terms of the Fibonacci sequence.
1
http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

29. Sunflower
Right spirals Left Spirals

30. Echinacea purpura

31. A
B
C
DNA
http://www.goldennumber.net/dna/

32. THE HUMAN BODY
http://jwilson.coe.uga.edu/emat6680/parveen/Fib_nature.htm
The ratio between the forearm and the hand is the Golden Ratio!

33. LEONARDO - THE VITRUVIAN MAN
http://en.wikipedia.org/wiki/Vitruvian_Man
The drawing is based on the correlations of ideal
human proportions with geometry described by the
ancient Roman architect Vitruvius in Book III of his
treatise De Architectura. Vitruvius described the
human figure as being the principal source of
proportion among the Classical orders of
architecture. Vitruvius determined that the ideal body
should be eight heads high. Leonardo’s drawing is
traditionally named in honor of the architect. This
image demonstrates the blend of art and science
during the Renaissance and provides the perfect
example of Leonardo’s deep understanding of
proportion.

34. SOLAR SYSTEM AND THE UNIVERSE
New findings reveal that the universe itself is in the shape
of a dodecahedron, a twelve-sided geometric solid
with pentagon faces, all based on phi.
http://www.goldennumber.net/cosmology/
Saturn’s magnificent rings show a division at a golden
section of the width of the rings
Curiously, even the relative distances of the ten planets and the largest asteroid average to phi

35. http://www.mnn.com/earth-matters/climate-weather/blogs/finding-the-fibonacci-sequence-in-hurricane-sandy
HURRICANE SANDY

36. SPIRAL GALAXY

37. ARCHITECTURE
NOTRE DAME CATHEDRAL

38. THE TAJ MAHAL
LE CORBUSIER

39. The Great Pyramid of
Egypt closely embodies
Golden Ratio
proportions.
http://www.goldennumber.net/phi-pi-great-pyramid-egypt/
There is debate as to the geometry used in the
design of the Great Pyramid of Giza in Egypt.
Built around 2560 BC, its once flat, smooth outer
shell is gone and all that remains is the roughly-
shaped inner core, so it is difficult to know with
certainty.
There is evidence, however, that the design of the
pyramid embodies these foundations of
mathematics and geometry:
•Phi, the Golden Ratio that appears throughout
nature.
•Pi, the circumference of a circle in relation to its
diameter.
•The Pythagorean Theorem – Credited by
tradition to mathematician Pythagoras (about 570
– 495 BC), which can be expressed as a² + b² =
c².
EGYPT

40. The CN Tower in Toronto, the tallest tower and
freestanding structure in the world, contains the
golden ratio in its design. The ratio of observation
deck at 342 meters to the total height of 553.33 is
0.618 or phi, the reciprocal of Phi!
John Hamilton Andrews, architect
Here is Plato’s divided line!

41. DESIGN
Upper case Phi is Φ and lower case phi is φ
http://www.goldennumber.net/logo-design/

42. FASHION CAR DESIGN
MEDICAL
HANDWRITING
http://jwilson.coe.uga.edu/emat6680/parveen/ubi_phi.htm
and http://www.goldennumber.net

43. Musical scales are based on Fibonacci numbers
Note how the piano keyboard of C to C above of 13 keys has 8 white keys and 5 black keys,
split into groups of 3 and 2
There are 13 notes in the span of any note through its octave.
A scale is comprised of 8 notes, of which the
5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is
2 steps from the root tone, that is the
1st note of the scale.
MUSIC
Here is a Fibonacci reminder
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377....etc.
http://www.goldennumber.net/music/

44. MOZART AND DEBUSSY BOTH USED THE GOLDEN RATIO IN THEIR
COMPOSITIONS, AS DID MANY OTHER COMPOSERS AND POETS
Musical instruments are often based on phi
http://jwilson.coe.uga.edu/emat6680/parveen/GR_in_art.htm

45. ART

46. Leonardo The Last Supper. Golden Sections all over the place!

47. Note on Da Vinci’s “The Annunciation” that the brick wall of the courtyard is at exact
golden ratio proportions in relation to the top and bottom of the painting see the diagram
on the next slide

48. I don’t know if Leonardo planned on all the Golden Ratios in this painting, but it is fun to see how many
associations we can fine. The Painting seems to be double Golden Rectangles, and that spiral joins the hand of the
angel with the hand of Mary perfectly!
http://abyss.uoregon.edu/~js/glossary/golden_rectangle.html

49. This self-portrait by Rembrandt (1606-1669) is an example of a triangular composition.
A perpendicular line from the apex of the triangle to the base cut the base in golden
section. Did he mean to do this?
http://jwilson.coe.uga.edu/emat6680/parveen/Golden_Triangle.htm

50. Michelangelo's Holy Family ... and Raphael Crucifixion are other examples, wherein the
principle figure outlines the Golden triangle which can be used to locate one of its
underlying pentagrams. The arms of the star form divided lines in Golden Ratios
PENTAGRAMS

51. Hans Hofmann, The Gate. 1959 Piet Mondrian, Composition in Red,
Yellow and Blue. 1935. $50,565,000 Mark Rothko, Violet, Green, Red.
1951

52. Matisse L’Escargot
Escargot is the French word for snail. Many mollusk shells follow some form of the Fibonacci sequence or are in
some way allied with the Golden Ratio. Here Is a golden spiral superimposed on top of Matisse’s artwork

53. Sacrament of the Last Supper by Salvador Dali
This is not “accidental”. Dali was fascinated by mathematics and geometry and
many of his monumental works are based on geometry and the Golden Ratio

54. M.C. ESCHER
Escher was one of the most famous of the mathematically inspired artists. His extraordinary tessellations,
Platonic Solids, polyhedrons, and other geometrically arranged shapes are fascinating to observe and in my
opinion he is not as well known as he should be.
Many of Escher's works contain impossible constructions, made using geometrical objects that cannot exist but
are pleasant to the human sight. Some of Escher's tessellation drawings were inspired by conversations with the
mathematician H. S. M. Coxeter concerning hyperbolic geometry. Relationships between the works of
mathematician Kurt Gödel, artist Escher, and composer Johann Sebastian Bach are explored in Gödel, Escher,
Bach, a Pulitzer Prize-winning book.
http://en.wikipedia.org/wiki/Mathematics_and_art#M.C._Escher

55. Henri Cartier-Bresson
Photography

56. Golden spiral Rule of thirds Infinity points

57. On the left is Monet’s Landscape at Vétheuil. Measure the short side of the canvas and flop it down to the long side
and you will have a square. That imaginary line is the Rabatment and it is used over and over by artists through the
centuries to design their canvases. Obviously when a canvas is all ready square finding the Rabatment is not possible.
So using the Rule of Thirds on Monet’s Houses of Parliament we find that he actually knew what he was doing!
Monet was well trained in classical techniques and these compositions are no accident.
RABATMENT RULE OF THIRDS

58. Renaissance Art Composition and
the Ukrainian Parliament Fight
August 7, 2014 by Gary Meisner
http://www.goldennumber.net/renaissance-art-composition-ukranian-parliament-
fight/
This photo and article was
all over the internet a month

60. Anatomy

61. The British contest in 2012 to find “Britain’s Perfect Face”

62. Thank you

63. RESOURCES
The Internet has hundreds, maybe more, websites and blogs with diagrams and photographs
about the Golden Ratio and Golden Rectangles. I have used many of them and I did not
remember to give every image a credit. There is so much online about the Golden Ratio and its
allied fields of mathematics it is overwhelming. I am not going to list all of them as a Google
search can bring all this wealth to your own computer. But here are a few I found useful….
www.goldennumber.net by Gary Meisner. Some of you might be interested in his software for
finding phi in your own compositions and photographs. Go to this website
www.phimatrix.com
http://jwilson.coe.uga.edu/emat6680/parveen/golden_rectangle.htm and many other articles are
here
http://www.museumofthegoldenratio.org/ Great information here as well as many links to art
related websites
http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm some interesting diagrams and
information on phi in nature
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
Most of the information was taken from: http://www.authorstream.com/Presentation/gerinzhills-
2279358-golden-ratio/

64. 1. http://en.wikipedia.org/wiki/Golden_ratio
2. http://en.wikipedia.org/wiki/List_of_works_designed_with_the_golden_ratio
3. http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature
4. http://www.livescience.com/37704-phi-golden-ratio.html
5. http://www.abc-people.com/data/leonardov/zolot_sech-txt.htm
6. http://russian7.ru/2014/09/zolotoe-sechenie-kak-ehto-rabotaet/
7. http://n-t.ru/tp/iz/zs.htm
8. http://ustierechi.ucoz.ru/publ/14-1-0-168
9. http://zhitanska.com/content/sakralnaya-geometriya
10. http://en.wikipedia.org/wiki/Fibonacci
11. https://ru.wikipedia.org/wiki/%D0%97%D0%BE%D0%BB%D0%BE%D1%82%D0%BE%D0%B5_%D1%81%D0%B5%D1%87%D0%B5%
D0%BD%D0%B8%D0%B5
12. https://en.wikipedia.org/wiki/Rule_of_thirds
13. https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0
%D1%87%D1%87%D0%B8
14. http://mathworld.wolfram.com/GoldenRatio.html
15. http://www.goldennumber.net/facial-beauty-golden-ratio-florence-colgate/
16. http://www.goldennumber.net/golden-ratio-myth/
17. http://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth#4
18. http://www.tutoronline.ru/blog/chisla-fibonachchi-ishhem-sekret-mirozdanija
19. !!! http://log-in.ru/articles/zolotoe-sechenie/
20. http://habrahabr.ru/post/154087/
21. !!! http://www.authorstream.com/Presentation/gerinzhills-2279358-golden-ratio/
22. !!! https://www.khanacademy.org/math/algebra/ratio-proportion-topic/advanced-ratios/v/the-golden-ratio
23. !!! http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html
24. !!! http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html