THE GOLDEN RATIO

Teresa Salazar Segura - 4ºA

ratio

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Introduction ............................................................................................................................. 3What is the golden ratio ....................................................................................................... 5Relation to the Fibonacci sequence ................................................................................ 8Phi number .............................................................................................................................. 11The golden mean ................................................................................................................... 15Conclusion .............................................................................................................................. 29

INDEX:

1. Introduction

i n t r o d u c t i o n

2. What is the golden ratio?

d e f i n i t i o n

The golden ratio is an irrational number that ancient thinkers discovered when they noticed the link between two segments belonging to the same line. This proportion can be found in nature and in geometric figures. It is given an aesthetic condition: that whose forms respect the golden ratio is considered beautiful.

golden number demonstration

3. Relation to the Fibonacci sequence

In mathematics, the Fibonacci sequence refers to the ordered sequence of numbers described by Leonardo of Pisa, an Italian mathematician. Each of the elements in the series is known as the Fibonacci number.

FIBONACCI SEQUENCE

If you take any number from the Fibonacci sequence and divide it by the previous number, we always get a very similar number. And this number to which all the results are approaching is called the golden number or Phi

How is the Fibonacci sequence related to the golden number?

4. Phi number

This number is a constant that represents the relationship between the sides of a rectangle.

PHI NUMBER

It is estimated that by the year 2000 B.C. it was the Babylonians who had the first approximations to the calculation of this number. They found that the circumference of a circle is usually a little more than three times, the equivalent of its diameter.

HISTORY OF PHI NUMBER

However, it was until 225 B.C. when Archimedes started with Mathematical Theory. This was perfected over the centuries.

The Englishman William Oughtred was the first to use the Greek letter π as a symbol of the quotient between the lengths of a circumference and its diameter.

5. The golden mean

The appearance of “The Golden Mean” in different brand logos

The golden ratio appears in the proportions that keep buildings, sculptures, objects, parts of our body, etc. An example of a golden rectangle in art is the elevation of the Greek Parthenon.

The importance of the "Golden Ratio" in a selection of works of art

In the world of architecture, the golden ratio is defined as the projection and design of direct mathematical proportions, where a specific object reaches within a geometric figure.

The importance of the "Golden Ratio" in a selection of architecture

Leonardo Fibonacci was the one who discovered, in the twelfth century, that a simple numerical series would be the basis of the relationship that we find behind the non-Phi golden number.

LEONARDO FIBONACCI

Proportion is the simplest statistical measure that can be assigned to a binary variable. The proportion expresses the frequency with which the variable of interest is found, within the total data set of a population. It is a parameter, and it is an exact number, that is, if the population does not change, the proportion will be constant.

Proportions of a group of people

The Modulor is a measurement tool derived from human stature and mathematics. It was developed as a visual bridge between two incompatible scales, the imperial system and the metric system.

Vitruvian modulor by Leonardo da Vinci

We say that the diagonals of a regular pentagon are in golden ratio with its sides. The point of intersection of two diagonals of a regular pentagon divides both in the golden ratio.

The divine proportion

Divine proportion in mosaics

• Although some planners or designers ignore the theme of the golden ratio, the reality is that throughout history it has been successfully applied in multiple projects, designs, buildings, photography, with mathematics playing an important role.

• Those illustrious artists and craftsmen understood that proportionality before the viewer is necessary to obtain a visual with harmony and composition, enhancing the beauty of things.

Golden ratio in trick

The Golden Ratio is a curious mathematical relationship present in nature: in the veins of the leaves, in the thickness of the branches, in the shell of mollusks, in the seeds of sunflowers, in the horns of goats, etc.

Golden ratio in nature

• Perhaps the most striking is the one related to the navel: if the total height of a man is divided by the distance from the navel to the feet, we obtain the golden number.

• It turns out that the quotient between the height of the man and the distance from the navel to the tip of the hand is the golden number.

Golden ratio in humans

• In fact, the role that the golden number plays in botany is so great that it is known as "Ludwig's Law"

• The golden number also relates to the number of male and female bees in a hive, or the arrangement of flower petals.

Golden ratio in botany

Golden ratio in painting

• Mathematics is about more than numbers and calculations.

• Beauty and perfection are implicit in the golden number.

• The golden ratio reminds us of the thought of Pythagoras: "at its deepest level reality is mathematical in nature"

C O N C L U S I O N

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