School of Scholars, Gadchiroli

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Presentation transcript:

School of Scholars, Gadchiroli presents……….. The Golden ratio A Geeky quest!!

Introduction : the golden ratio Origins with the Fibonacci sequence Discovered by Leonardo Fibonacci Born- 1175 AD, Pisa (Italy) Died – 1250 AD

The Fibonacci series begins with 0,1 add the previous 2 numbers to get the next one Let’s go : 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610……………

Relationship between the Fibonacci series and golden ratio Divide the next number by the previous number Let’s start from 1 After some time, the division has 1.618 as its beginning numbers. 2/1 = 2.0 3/2 = 1.5 5/3 = 1.67 8/5 = 1.6 13/8 = 1.625 21/13 = 1.615 34/21 = 1.619 55/34 = 1.618 89/55 = 1.618

The Golden Section

The Golden Ratio, Phi 1.618033989………………….. An irrational number also known as Phi (φ) which is approximately equal to 1.618o33989…………………… What makes it so special then?

Reciprocal of the golden ratio The reciprocal of golden ratio, 1/1.618……. approximately equals 0.618. Golden ratio is a number which is equal to its own reciprocal plus 1. Φ = _1 + 1 Φ Now let’s see the solutions for golden ratio.

The solution The equation can be written as φ2 - φ - 1 = 0, So, we can derive the value of the Golden Ratio from the quadratic equation, x = -b  √b²-4ac 2a with a = 1, b = -1, and c = -1: Φ = 1  √5 ≈ 1.618033989…………………………… 2 Negative sign’s not used because a negation yeilds the negative reciprocal of the golden ratio, i.e. -0.618 033 989

The golden Ratio : an absurd! The golden ratio is quite absurd; It’s not your ordinary surd. If you invert it (this is fun!), You’ll get itself, reduced by one; But if you increase it by one, We get its square! We intend no pun! Fibonacci Quarterly, 1970 --> Get the exact reference. (3LB -- QA241 .A1 F52)

Infinite square roots

Infinite continued fraction Expressed as a continued fraction, It’s one, one, one, …, until distraction; In short, the simplest of such kind (Doesn’t this really blow your mind?)

Some researches! Value of √5 √5 = 2Φ – 1 = 2 (1.618) – 1 = 3.236 – 1 ≈ 2.236 2Φ - √5 = 1 To check eye faults

Assumptions for the golden ratio A sacred ratio used in building Egyptian pyramids 2,600 years ago Mentioned in the Ahmes Papyrus, a 17th century book of math Eudoxus of Cnidus (c. 370 B.C.) observed that his friends divided a stick into golden proportions when asked to find the most pleasing placement of a crossbar.

Rectangle problem Which, according to you is the most attractive one? 1 2 1.618 1 1 1 Which, according to you is the most attractive one? Ratio of the sides of the 3 rectangles=1:1 2:1 1.618:1

Why so? The golden rectangle is the most appealing one. That’s because, naturally the golden ratio is the most appealing number in the universe That’s why we see things arranged in golden numbers (Fibonacci numbers) Let’s study this by natural and artificial objects!

A Golden Tree!

The Golden Spiral

Examples of the Golden Spiral in Nature....

Examples of the Golden Spiral in Nature....

Examples of the Golden Spiral in Nature....

Phyllotaxy

Your Beauty What has Φ got to do with beauty???? In fact, it has a lot to do!!!!

Let’s create a golden rectangle

Golden Ratio in architecture

Why? So, why do things that exhibit in the golden ratio seem so perfect and more appealing to the human eye? Why is it named the golden ratio and not silver or bronze?

Golden Mean Gauge: Invented by Dr. Eddy Levin DDS

The golden ratio, do you have it!

References Foremost, the nature, of course! www.youtube.com www.scribd.com www.freemansory.com www.suffolkmaths.com

Researched by: Mst. Ankush Raut, IX Ms. Roshni Gota, VIII Ms. Vaidehi Meshram, VII