GOLDEN RATIO

MADE BY DIVYANSH MISHRA AND

VARUN CHOPRA

WHAT IS GOLDEN RATIO ? DIVYANSH

• In mathematics, two quantities are in the golden ratio if In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for of the two quantities. Expressed algebraically, for quantities a and b with a > b,quantities a and b with a > b,

• where the Greek letter phi ( ) represents the golden ratio. where the Greek letter phi ( ) represents the golden ratio. Its value isIts value is

• ::• The golden ratio is also called the golden The golden ratio is also called the golden

section (Latin: section aurea) or golden mean. Other names section (Latin: section aurea) or golden mean. Other names include extreme and mean ratio,medial section, divine include extreme and mean ratio,medial section, divine proportion, divine section (Latin: section divina), golden proportion, divine section (Latin: section divina), golden proportion, golden cut, and golden number.proportion, golden cut, and golden number.

• Many artists and architects have proportioned their Many artists and architects have proportioned their works to approximate the golden ratio—especially in works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing believing this proportion to be aesthetically pleasing (see Applications and observations (see Applications and observations below). Mathematicians since Euclid have studied the below). Mathematicians since Euclid have studied the properties of the golden ratio, including its properties of the golden ratio, including its appearance in the dimensions of a regular pentagon appearance in the dimensions of a regular pentagon and in a golden rectangle, which can be cut into a and in a golden rectangle, which can be cut into a square and a smaller rectangle with the same aspect square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-the proportions of natural objects as well as man-made systems such as financial markets, in some made systems such as financial markets, in some cases based on dubious fits to data.cases based on dubious fits to data.

• At least 1,000,000,000,000 decimal digits are knownAt least 1,000,000,000,000 decimal digits are known

CALCULATIONLCULATION • Two quantities a and b are said to be in golden ratio Two quantities a and b are said to be in golden ratio

ifif• A+b/a = a/b =φA+b/a = a/b =φ• One method for finding the value of φ is to start with One method for finding the value of φ is to start with

the left fraction. Through simplifying the fraction the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,and substituting in b/a = 1/φ,

• A+b/a=1+b/a= 1+1/φA+b/a=1+b/a= 1+1/φ• By definition, it is shown thatBy definition, it is shown that• 1+1/φ =φ1+1/φ =φ• Multiplying by Multiplying by φφ givesgives

ΦΦ + 1= + 1= φ φ . . ΦΦwhich can be rearranged towhich can be rearranged to

ΦΦ . . ΦΦ – – Φ Φ-1 =0-1 =0Using the quadratic formula, two solutions Using the quadratic formula, two solutions

are obtained:are obtained:

ΦΦ = 1+root5 / 2 = 1.61803………… = 1+root5 / 2 = 1.61803…………AndAndΦΦ = 1 – root5/2 =.61803…………….. = 1 – root5/2 =.61803……………..Because Because φφ is the ratio between positive  is the ratio between positive

quantities quantities φφ is necessarily positive is necessarily positive

ΦΦ = 1+root5 / 2 = 1.61803………… = 1+root5 / 2 = 1.61803…………

Uses in architecture date to the ancient Egyptians

and Greeks• It appears that the Egyptians may have used both pi appears that the Egyptians may have used both pi

and phi in the design of the Great Pyramids. The and phi in the design of the Great Pyramids. The Greeks are thought by some to have based the Greeks are thought by some to have based the design of the Parthenon on this proportion, but this is design of the Parthenon on this proportion, but this is subject to some conjecture.subject to some conjecture.

• Phidias (500 BC – 432 BC), a Greek sculptor and Phidias (500 BC – 432 BC), a Greek sculptor and mathematician, studied phi and applied it to the mathematician, studied phi and applied it to the design of sculptures for the Parthenon.design of sculptures for the Parthenon.

• Plato (circa 428 BC – 347 BC), in his views on natural Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding considered the golden section to be the most binding of all mathematical relationships and the key to the of all mathematical relationships and the key to the physics of the cosmos.physics of the cosmos.

• Euclid (365 BC – 300 BC), in “Elements,” Euclid (365 BC – 300 BC), in “Elements,” referred to dividing a line at the referred to dividing a line at the 0.6180399… point as “dividing a line in 0.6180399… point as “dividing a line in the extreme and mean ratio.” This later the extreme and mean ratio.” This later gave rise to the use of the term mean in gave rise to the use of the term mean in the golden mean.  He also linked this the golden mean.  He also linked this number to the construction of a number to the construction of a pentagrampentagram.

The Fibonacci Series was discovered around 1200 AD

• Leonardo Fibonacci, an Italian born in 1175 Leonardo Fibonacci, an Italian born in 1175 AD (2) discovered the unusual properties of AD (2) discovered the unusual properties of the numerical series that now bears his the numerical series that now bears his name, but it’s not certain that he even name, but it’s not certain that he even realized its connection to phi and the Golden realized its connection to phi and the Golden Mean. His most notable contribution to Mean. His most notable contribution to mathematics was a work known as Liber mathematics was a work known as Liber Abaci, which became a pivotal influence in Abaci, which became a pivotal influence in adoption by the Europeans of the Arabic adoption by the Europeans of the Arabic decimal system of counting over Roman decimal system of counting over Roman numerals. (3)numerals. (3)

Golden proportion was first called the “Divine

Proportion” in the 1500′s• Da Vinci provided illustrations for a dissertation Da Vinci provided illustrations for a dissertation

published by Luca Pacioli in 1509 entitled “De Divina published by Luca Pacioli in 1509 entitled “De Divina Proportione” (1), perhaps the earliest reference in Proportione” (1), perhaps the earliest reference in literature to another of its names, the “Divine literature to another of its names, the “Divine Proportion.”  This book contains drawings made by Proportion.”  This book contains drawings made by Leonardo da Vinci of the five Platonic solids.  It was Leonardo da Vinci of the five Platonic solids.  It was probably da Vinci who first called it the “sectio aurea,” probably da Vinci who first called it the “sectio aurea,” which is Latin for golden sectionwhich is Latin for golden section.

• The Renaissance artists used the Golden Mean The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve extensively in their paintings and sculptures to achieve balance and beauty. Leonardo Da Vinci, for instance, balance and beauty. Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his used it to define all the fundamental proportions of his painting of “The Last Supper,” from the dimensions of painting of “The Last Supper,” from the dimensions of the table at which Christ and the disciples sat to the the table at which Christ and the disciples sat to the proportions of the walls and windows in the background.proportions of the walls and windows in the background.

• Johannes Kepler (1571-1630), discoverer of Johannes Kepler (1571-1630), discoverer of the elliptical nature of the orbits of the the elliptical nature of the orbits of the planets around the sun, also made mention planets around the sun, also made mention of the “Divine Proportion,” saying this of the “Divine Proportion,” saying this about it:about it:

• ““Geometry has two great treasures: one is Geometry has two great treasures: one is the theorem of Pythagoras; the other, the the theorem of Pythagoras; the other, the division of a line into extreme and mean division of a line into extreme and mean ratio. The first we may compare to a ratio. The first we may compare to a measure of gold; the second we may name measure of gold; the second we may name a precious jewel.”a precious jewel.”

The term “Phi” was not used until the 1900′s

• It wasn’t until the 1900′s that American mathematician Mark Barr used the Greek letter phi (Φ) to designate this proportion. By this time this ubiquitous proportion was known as the golden mean, golden section and golden ratio as well as the Divine proportion.  Phi is the first letter of Phidias (1), who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter “F,” the first letter of Fibonacci.  Phi is also the 21st letter of the Greek alphabet, and 21 is one of numbers in the Fibonacci series.  The character for phi also has some interesting theological implications.

Recent appearances of Phi in math and physics

• Phi continues to open new doors in Phi continues to open new doors in our understanding of life and the our understanding of life and the universe.  It appeared in Roger universe.  It appeared in Roger Penrose’s discovery in the 1970′s of Penrose’s discovery in the 1970′s of “Penrose Tiles,” which first allowed “Penrose Tiles,” which first allowed surfaces to be tiled in five-fold surfaces to be tiled in five-fold symmetry.  It appeared again in the symmetry.  It appeared again in the 1980′s in quasi-crystals, a newly 1980′s in quasi-crystals, a newly discovered form of matterdiscovered form of matter.

MATERIAL CONTRIBUTED BY OTHER GROUP MATES

AND FOR SOME OF YOU WHO DID NOT WATCH THE PPT , THE PPT WAS

MATHEMATICIANS OF WHICH COUNTRY FIRST DEVELOPED BASIC

NUMERICALS

WHY NOBEL PRIZE IS NOT GIVEN FOR MATHS

WHICH IS THE HIGHEST AWARD FOR MATHS

FIELDS MEDAL

NAME OF MATHS IN HINDI

HOW MANY DECIMAL DIGITS OF GOLDEN RATIO ARE THERE

NAME OF SIGN OF GOLDEN RATIO

GOLDEN RATIO IS USED IN WHICH SCIENCE(SCEINCE + MATHS)

HINT: GREEKS AND EGYPTIAN WERTE FIRST TO USE IT

WHO DISCOVERED PIE

PYTHOGORAS BELONGED TO WHICH CITY

• SAMON