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THE GOLDEN MEAN AND ITS APPLICATIONS

GOLDEN MEAN

• Phi (Golden Ratio) as a mysterious number has been discovered in many places, such as art, architectures, humans, and plants.

• According to the history of mathematics, Phi was first understood and used by the ancient mathematicians in Egypt, two to three thousand years ago, due to its frequent appearance in Geometry.

THE HISTORY OF THE GOLDEN RATIO

CONTD..

• The name "Golden Ratio" appears in the form sectio aurea (Golden Section in Greek) by Leonardo da Vinci who used this the Golden ratio in many of his masterpieces, such as The Last Supper and Mona Lisa.

• In 1900s, an American mathematician named Mark Barr, represented the Golden Ratio by using a greek symbol Φ.

Why are the objects that contain the

GOLDEN RATIO so pleasing?

THE SECRET OF THE GOLDEN RATIO

• Objects that meet the requirements of the Golden Ratio are attractive and pleasing to the human eye.

• Throughout history, many experts have tried to find reasonable explanations for this question. Two thousand years ago, ancient Greeks discovered the magic of this ratio from the Golden Rectangle.

• The Golden Rectangle is a rectangle that meets the Golden Ratio requirements and contains an infinite number of proportional Golden Rectangles within itself.

• The ancient Greeks were attracted by this special number and its unique characteristics.

MATHEMATICAL PROPERTIES OF THE GOLDEN RATIO

The Golden Ratio - Φ is an irrational numberthat has the following unique properties:

1. Taking the reciprocal of Φ and adding one yields Φ. phi=(1/phi)+1, or Φ = (1/Φ) + 1.

2. Φ squared equals itself plus one. Φ^2=Φ+1. it is the only number in the world that has such properties.

3. If we convert the equation from 2 into the equation Φ^2-Φ-1=0. Doing this we get x = (1 ± √5)/2. Together, these two solutions are known as Phi (1.618033989) and phi (0.618033989). Phi and phi are reciprocals.

Φ vs. Π

Φ and Π (pi) have this in common: where Π is the ratio of the circumference to its diameter, Φ is the ratio of the length to the width of a perfect rectangle.

WHAT IS THE GOLDEN RATIO?

The Golden Ratio is a unique number, approximately 1.618033989. It is also known as the Divine Ratio, the Golden Mean, the Golden Number, and the Golden Section.

WHAT IS THE FIBONACCI SEQUENCE OF NUMBERS?

The Fibonacci numbers are a unique sequence of integers, starting with 1, where each element is the sum of the two previous numbers. For example: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.

RELATIONSHIP BETWEEN THE FIBONACCI SEQUENCE AND THE GOLDEN RATIO

• The Fibonacci Sequence is an infinite sequence, which means it goes on for ever, and as it develops, the ratio of the consecutive terms converges (becomes closer) to the Golden Ratio, ~1.618.

• For example, to find the ratio of any two successive numbers, take the latter number and divide by the former. So, we will have: 1/1=1, 2/1=2, 3/2=1.5, 5/3=1.66, 8/5=1.6, 13/8=1.625, 21/13=1.615.

As we can see, the ratio approaches the Golden Ratio, which is also known as the Golden Section and Golden Mean. Even though we know it approaches this one particular constant, we can see from the graph that it will never reach this exact value.

• The Golden Ratio was originally derived from the golden rectangle. The following is one method to construct a golden rectangle:

CONSTRUCTING A GOLDEN RECTANGLE

Given: a square ABCD1.Find midpoint on DC.2.Connect MB.3.Draw a circle with the center of M, radius of MB.4.Expand the DC until it meets with the circle. The intersection is one vertex of the rectangle.5.Complete the rectangle.

During the whole process, we have made the two exactly proportional rectangles, each with the side ratio Φ, the Golden Ratio.

CONTD…• The Golden Rectangle is said to be one of the most

visually satisfying of all geometric forms; whose length: width ratio is equal to Phi.

THE GOLDEN SPIRAL

• When a Golden Rectangle is progressively subdivided into smaller and smaller Golden Rectangles the pattern below is obtained. From this, a spiral can be drawn which grows logarithmically, where the radius of the spiral, at any given point, is the length of the corresponding square to a Golden Rectangle. This is called the Golden Spiral.

The Human Face• Dr. Stephen Marquardt a former plastic

surgeon, used the golden section and some of its relatives to make a mask that he claims that is the most beautiful shape a human face can ever have, it used decagons and pentagons as its function that embodies phi in all their dimensions. Dr. Marquardt has been studying on human's facial beauty for many years.

THE GOLDEN RATIO AND BEAUTY IN HUMANS

THE HUMAN SMILE

A perfect smile: the front two teeth form a golden rectangle (which is said to be one of the most visually satisfying of forms, as it is formed with sides of 1 and 1.618). There is also a Golden Ratio in the height to width of the center two teeth. And the ratio of the width of the two center teeth to those next to them is phi. And, the ratio of the width of the smile to the third tooth from the center is also phi.

THE HUMAN LUNGS

The windpipe divides into two main bronchi, one long (the left) and the other short (the right). This asymmetrical division continues into the subsequent subdivisions of the bronchi. It was determined that in all these divisions the proportion of the short bronchus to the long was always 1:1.618.

WHAT KIND OF RELATIONSHIP DO

FIBONACCI NUMBERS AND PLANTS HAVE?

The process of the growing plant follows the Fibonacci numbers, from the first shoot, to the two shoots, three shoots, and five shoots, and eight shoots, and on and on.The branching rates in plants occur in the Fibonacci pattern, where the first level has one "branching" (the trunk), the second has two branches, than 3, 5, 8, 13 and so on. Also, the spacing of leaves around each branch or stalk spirals with respect to the Golden Ratio.

THE GOLDEN RATIO AND BEAUTY IN NATURE

Plant Growth

HAVE YOU EVER WONDERED WHY

MONA LISA PAINTING IS SO BEAUTIFUL?

• The Golden Ratio has a great impact on art, influencing artists' perspectives of a pleasant art piece. Mona Lisa's face is a perfect golden rectangle, according to the ratio of the width of her forehead compared to the length from the top of her head to her chin.

THE GOLDEN RATIO AND BEAUTY IN ART

• The Golden Ratio has appeared in ancient architecture. The examples are many, such as the Great Pyramid in Giza, Egypt which is considered one of the Seven World Wonders of the ancient world, and the Greek Parthenon that was constructed between 447 and 472BC.

THE GOLDEN RATIO AND BEAUTY IN ARCHITECTURE

The Great Pyramid at Giza

• Half of the base, the slant height, and the height from the vertex to the center create a right triangle. When that half of the base equal to one, the slant height would equal to the value of Phi and the height would equal to the square root of Phi.

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

GOLDEN RATIO IN MUSIC

MUSICAL SCALES ARE BASED ON FIBONACCI NUMBERS

• There are 13 notes in the span of any note through its octave.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

CONCLUSION:

• An interesting result has occurred due to our research, we now see the examples of the Golden Ratio everywhere. It is as if our eyes have been opened to something that existed all around us but to see. For that we are thankful.