http://goldennumber.net/dna.htm

Musical scales are based on Fibonacci numbers http://jwilson.coe.uga.edu/EMAT6680/Parveen/GR_in_art.htm

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.

Note too 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

Another aspect of the golden ratio in music is illustrated in compositions by Mozart. Mozart's piano sonatas

have been observed to display use of the golden ratio through the arrangement of sections of measures that

make up the whole of the piece. In Mozart's time, piano sonatas were made up of two sections, the

exposition and the recapitulation. In a one hundred measure composition it has been noted that Mozart

divided the sections between the thirty-eighth and sixty-second measures. This is the closest approximation

that can be made to the Golden Ratio within the confines of a one-hundred measure composition. Some

scholars have debunked this theory since further analysis of such compositions have shown that the Golden

Ratio was not consistently applied within the subsections of the same compositions. Others state that this

does not prove that he did not utilize the Golden Ratio, only that he did not apply it to all aspects of

particular compositions. Whether he applied the Golden Ratio intentionally or used it intuitively is not

known but studies seem to indicate the latter.

The Golden Ratio has also appeared in poetry in much the same way that it appears in music. The emphasis

has been placed on time intervals. Some have even stated that the meaning of chosen words is less

important than its rhythmic quality and the intervals between words and lines that serve to create the overall

rhythm of a poem.

Probably the most compelling display of the Golden Ratio is in the many examples seen in nature. The

Golden Ratio and the Fibonacci Sequence can be seen in objects from the human body to the growth

pattern of a chambered nautilus. Examples of the Fibonacci Sequence can be seen in the growth pattern of a

tree branch or the packing pattern of seeds on a flower. Ultimately, this aspect is what has earned the

Golden Mean its representation as the Divine Proportion

It is the prevalence of the Golden Ratio in nature that has influenced classic art and architecture. The great

masters developed their skills by recreating things they observed in nature. In the earliest of cases, these

artists and craftsmen probably had no knowledge of the math involved, only an acute awareness of this

pattern repeated around them. It was the mathematicians that unlocked the secrets of the Golden Ratio.

Their work has led to the understanding of the complex mathematical underpinnings hidden within the

Golden Mean.

Musical instruments are often based on phi

Fibonacci and phi are used in the design of violins and even in the design of high quality speaker wire.

Golden Ratio in Art and Architecture

By Samuel Obara http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.htm

l

I. A discovery of Golden Ratio A. A brief history of Golden Ratio

There are many different names for the golden ratio; The Golden Mean, Phi, the Divine

Section, The Golden Cut, The Golden Proportion, The Divine Proportion, and tau(t).

The Great Pyramid of Giza built around 2560 BC is one of the earliest examples of the

use of the golden ratio. The length of each side of the base is 756 feet, and the height is

481 feet. So, we can find that the ratio of the vase to height is 756/481=1.5717.. The

Rhind Papyrus of about 1650 BC includes the solution to some problems about pyramids,

but it does not mention anything about the golden ratio Phi.

Euclid (365BC - 300BC) in his "Elements" calls dividing a line at the 0.6180399.. point

dividing a line in the extreme and mean ratio. This later gave rise to the name Golden

Mean. He used this phrase to mean the ratio of the smaller part of this line, GB to the

larger part AG (GB/AG) is the same as the ratio of the larger part, AG, to the whole line

AB (AG/AB).Then the definition means that GB/AG = AG/AB.

proposition 30 in book VI

Plato, a Greek philosopher theorised about the Golden Ratio. He believed that if a line

was divided into two unequal segments so that the smaller segment was related to the

larger in the same way that the larger segment was related to the whole, what would

result would be a special proportional relationship.

Luca Pacioli wrote a book called De Divina Proportione (The Divine Proportion) in 1509.

It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. Leonardo Da

Vinci first called it the sectio aurea (Latin for the golden section).

Today, mathematicians also use the initial letter of the Greek Phidias who used the

golden ratio in his sculptures.

B. Definitions of Golden Ratio

1) Numeric definition

Here is a 'Fibonacci series'.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..

If we take the ratio of two successive numbers in this series and divide each by the

number before it, we will find the following series of numbers.

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.6666...

8/5 = 1.6

13/8 = 1.625

21/13 = 1.61538...

34/21 = 1.61904...

The ratio seems to be settling down to a particular value, which we call the golden

ratio(Phi=1.618..).

2) Geometric definition

We can notice if we have a 1 by 1 square and add a square with side lengths equal to the

length longer rectangle side, then what remains is another golden rectangle. This could go

on forever. We can get bigger and bigger golden rectangles, adding off these big squares.

Step 1 Start with a square 1 by 1

Step 2 Find the longer side

Step 3 Add another square of that side to whole thing

Here is the list we can get adding the square;

1 x 1, 2 x 1, 3 x 2, 5 x 3, 8 x 5, 13 x 8, 21 x 13, 34 x 21.

with each addition coming ever closer to multiplying by Phi.

start 1 by 1, add 1 by 1 => Now, it is 2 by 1, add 2 by 2

Now, it is 3 by 2, add 3 by 3 => Now, it is 5 by 3, add 5 by 5

Now, it is 8 by 5.

3) Algebraic and Geometric definition

We can realize that Phi + 1 = Phi * Phi.

Start with a golden rectangle with a short side one unit long.

Since the long side of a golden rectangle equals the short side multiplied by Phi, the long

side of the new rectangle is 1*Phi = Phi.

If we swing the long side to make a new golden rectangle, the short side of the new

rectangle is Phi and the long side is Phi * Phi.

We also know from simple geometry that the new long side equals the sum of the two

sides of the original rectangle, or Phi + 1. (figure in page4)

Since these two expressions describe the same thing, they are equivalent, and so

Phi + 1 = Phi * Phi.

II. Some Golden Geometry

1) The Golden Rectangle

A Golden Rectangle is a rectangle with proportions that are two consecutive numbers

from the Fibonacci sequence.

The Golden Rectangle has been said to be one of the most visually satisfying of all

geometric forms. We can find many examples in art masterpieces such as in edifices of

ancient Greece.

2) The Golden Triangle

If we rotate the shorter side through the base angle until it touches one of the legs, and

then, from the endpoint, we draw a segment down to the opposite base vertex, the

original isosceles triangle is split into two golden triangles. Aslo, we can find that the

ratio of the area of the taller triangle to that of the smaller triangle is also 1.618. (=Phi)

If the golden rectangle is split into two triangles, they are called golden triangles suing

the Pythagorean theorem, we can find the hypotenuse of the triangle.

3) The Golden Spiral

The Golden Spiral above is created by making adjacent squares of Fibonacci dimensions

and is based on the pattern of squares that can be constructed with the golden rectangle.

If you take one point, and then a second point one-quarter of a turn away from it, the

second point is Phi times farther from the center than the first point. The spiral increases

by a factor of Phi.

This shape is found in many shells, particularly the nautilus.

4) Penrose Tilings

The British physicist and mathematician, Roger Penrose, has developed an aperiodic

tiling which incorporates the golden section. The tiling is comprised of two rhombi, one

with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to

base) and one with angles of 72 and 108 degrees (figure B).

When a plane is tiled according to Penrose's directions, the ratio of tile A to tile B is the

Golden Ratio.

In addition to the unusual symmetry, Penrose tilings reveal a pattern of overlapping

decagons. Each tile within the pattern is contained within one of two types of decagons,

and the ratio of the decagon populations is, of course, the ratio of the Golden Mean.

5) Pentagon and Pentagram

We can see there are lots of lines divided in the golden ratio. Such lines appear in the

pentagon and the relationship between its sides and the diagonals.

We can get an approximate pentagon and pentagram using the Fibonacci numbers as

lengths of lines. In above figure, there are the Fibonacci numbers; 2, 3, 5, 8. The ratio of

these three pairs of consecutive Fibonacci numbers is roughly equal to the golden ratio.

III. Golden Ratio in Art and Architecture

A. Golden Ratio in Art

1) An Old man by Leonardo Da Vinci

Leonardo Da Vinci explored the human body involving in the ratios of the lengths of

various body parts. He called this ratio the "divine proportion" and featured it in many of

his paintings.

Leonardo da Vinci's drawing of an old man can be overlaid with a square subdivided into

rectangles, some of which approximate Golden Rectangles.

2) The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci

We can draw many lines of the rectangles into this figure.

Then, there are three distinct sets of Golden Rectangles: Each one set for the head area,

the torso, and the legs.

3) Mona-Risa by Leonardo Da Vinci

This picture includes lots of Golden Rectangles. In above figure, we can draw a rectangle

whose base extends from the woman's right wrist to her left elbow and extend the

rectangle vertically until it reaches the very top of her head. Then we will have a golden

rectangle.

Also, if we draw squares inside this Golden Rectangle, we will discover that the edges of

these new squares come to all the important focal points of the woman: her chin, her eye,

her nose, and the upturned corner of her mysterious mouth.

It is believed that Leonardo, as a mathematician tried to incorporate of mathematics into

art. This painting seems to be made purposefully line up with golden rectangle.

4) Holy Family by Micahelangelo

We can notice that this picture is positioned to the principal figures in alignment with a

Pentagram or Golden star.

5) Crucifixion by Raphael

his picture is a well-known example, in which we can find a Golden Triangle and also

Pentagram. In this picture, a golden triangle can be used to locate one of its underlying

pentagrams.

6) self-portrait by Rembrandt

We can draw three straight lines into this figure. Then, the image of the feature is

included into a triangle. Moreover, if a perpendicular line would be dropped from the

apex of the triangle to the base, the triangle would cut the base in Golden Section.

7) The sacrament of the Last Supper by Salvador Dali(1904-1989)

This picture is painted inside a Golden Rectangle. Also, we can find part of an enormous

dodecahedron above the table. Since the polyhedron consists of 12 regular Pentagons, it

is closely connected to the golden section.

8) Golden Section Plate 1, 1993 by Fletcher Cox

The title of this work itself includes the Golden Section. It simply means that it is cut into

sections of Golden Proportion.

9) Bathers by Seurat

Seurat attached most of canvas by the Golden Section. This picture has several golden

subdivisions.

10) Composition with Gray and Light Brown by Piet Mondrian 1918

Mondrian believed that mathematics and art were closely connected. He used the

simplest geometrical shapes and primary colours (blue, red, yellow).

His point of view lies in the fact that any shape is possible to create with basic geometric

shapes as well as any color can be created with different combinations of red, blue, and

yellow. The golden rectangle is one of the basic shapes appear in Mondrian's art.

Composition in Red, Yellow, and Blue(1926)

We can find that the ratio of length to width for some rectangles is Phi.

B. Golden Ratio in Architecture

1) The Great Pyramid

The Ahmes papyrus of Egypt gives an account of the building of the Great Pyramid of

Giaz in 4700 B.C. with proportions according to a "sacred ratio."

2) Parthenon

The Greek sculptor Phidias sculpted many things including the bands of sculpture that

run above the columns of the Parthenon.

Even from the time of the Greeks, a rectangle whose sides are in the "golden proportion"

has been known since it occurs naturally in some of the proportions of the Five Platonic.

This rectangle is supposed to appear in many of the proportions of that famous ancient

Greek temple in the Acropolis in Athens, Greece.

3) Porch of Maidens, Acropolis, Athens

4) Chartres Cathedral

The Medieval builders of churches and cathedrals approached the design of their

buildings in much the same way as the Greeks. They tried to connect geometry and art.

Inside and out, their building were intricate construction based on the golden section.

5) Le Corbussier

In 1950, the architect Le Corbussier published a book entitled "Le modulator. Essai

sur une mesure harmonique a l'echelle humaine applicable universalement a l'architecture

et a la mecanique ". He invented the word "modulator" by combining "modul" (ratio) and

"or" (gold); another expression for the well-known golden ratio.

III. Resoureces

Internet

Michael's Crazy Enterprises, Inc., The Golden Mean

(http://www.vashti.net/mceinc/)

The Golden Ratio

(http://www.math.csusb.edu/course/m128/golden/)

Ron Knott, The Golden section ratio : Phi

(http://www.ee.surrey.ac.uk/Personal/R.Knott/)

The Golden Ratio

(http://library.thinkquest.org/C005c449/)

Ron Knott, Fibonacci Numbers and Nature-part 2, Why is the Golden section the "best"

arrangement?

(http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/)

Ron Knott, The Golden Section in Art, Architecture and Music

(http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/)

Steve Blacker, Jeantte Polanski, and Marc Schwach, The Golden Ratio

(http://www.geom.umn.edu/~demo5337/s97b/)

Ethan, The relations of the Golden ratio and the Fibonacci Series

(http://mathforum.org/dr.math/problems)

Golden Section in Art and Architecture

(http://www.camosun.bc.ca/~jbritton/goldslide/)

Sheri Davis and Danny Rhee, Mathematical Aspects of Arichitecture

(http://www.ma.uyexas.edu/~lefcourt/SP97/M302/projects/lefc023/)

Mathematics and Art

(http://www.q-net.au/~lolita/)

Leonardo da Vinci

(http://libray.thinkquest.org/27890/)

Math & Art : The golden Rectangle

(http://educ.queensu.ca/~fmc/october2001/)

Sue Meredith, Some Explorations with the Golden Ratio

( http://jwilson.coe.uga.edu/EMT668/)

What is a Fractal?

(http://ecsd2.re50j.k12.co.us/ECSD/)

Ron Knott, Phi's Fascinating Figures

(http://www.euler.slu.edu/teachmaterial/)

Cynthia Lanius, Golden ratio Algebra

(http://math.rice.edu/~lanius/)

Newsletter, Mathematical Beauty

(http://www.exploremath.com/news/

Some Golden Geometry

(http://galaxy.cau.edu/tsmith/KW/)

Book Robert L. (1989). Scared goemetry: philosophy and practice, New York: Thames

and Hudson.

Article

Donald, T. S. (1986). The Geometric Figure Relating the Golden Ratio and Phi,

Mathematics Teacher 79, 340-341.

Edwin, M. D. (1993). The Golden Ratio: A good opportunity to investigate multiple

representations of a problem, Mathematics Teacher 86, 554-557.

Susan, M. P. (1982). The Golden Ratio in Geometry, E. M. Maletsky, C. Hirsch, & D.

Yates(Eds.), Mathematics Teacher 75, 672-676.