Ruhulla Abdul IFF 1-2 students

Moscow 2015

The Golden Ratio

Financial University under the Government of the Russian Federation

SO WHAT IS A RATIO?Remember this?

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.

the ratio of the length of the entire line (A)to the length of larger line segment (B)

is the same asthe 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:

           

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

Phi 1.6180339887…..the Golden Number!

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

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

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

How to make a Golden Rectangle without math

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!

PUTTING THEM ALL BACK TOGETHER

HERE IS A GOLDEN SPIRAL

EACH LINE IS ALSO A GOLDEN RATIO

HERE IS THE INFINITY POINT

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!

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

Ф in pentagram

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

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…

The Golden Spiral constructed from a Golden Rectangle

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.

PETAL ARRANGEMENTS

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

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

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

Sunflower 

Right spirals Left Spirals

Echinacea purpura

A

B

C

DNA

http://www.goldennumber.net/dna/

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!

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.

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

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

HURRICANE SANDY

SPIRAL GALAXY

ARCHITECTURE

NOTRE DAME CATHEDRAL

THE TAJ MAHALLE CORBUSIER

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

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!

DESIGN

Upper case Phi is  Φ and lower case phi is φ

http://www.goldennumber.net/logo-design/

FASHION CAR DESIGN

MEDICAL

HANDWRITING

http://jwilson.coe.uga.edu/emat6680/parveen/ubi_phi.htmand http://www.goldennumber.net

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 the5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is2 steps from the root tone, that is the1st note of the scale.

MUSIC

Here is a Fibonacci reminder0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377....etc.

 

http://www.goldennumber.net/music/

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

ART

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

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

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

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

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

Hans Hofmann, The Gate. 1959  Piet Mondrian, Composition in Red, Yellow and Blue. 1935. $50,565,000

 

Mark Rothko, Violet, Green, Red. 1951

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

Sacrament of the Last Supper by Salvador Dali

This is not “accidental”. Dali was fascinated by mathematics and geometry andmany of his monumental works are based on geometry and the Golden Ratio

M.C. ESCHEREscher 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

Henri Cartier-Bresson

Photography

Golden spiral Rule of thirds Infinity points

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

Renaissance Art Composition and the Ukrainian Parliament FightAugust 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

Anatomy

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

Thank you

RESOURCESThe 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 websitewww.phimatrix.comhttp://jwilson.coe.uga.edu/emat6680/parveen/golden_rectangle.htm and many other articles are herehttp://www.museumofthegoldenratio.org/ Great information here as well as many links to art related websiteshttp://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm some interesting diagrams and information on phi in naturehttp://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/

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