THE GOLDEN RATIO also known as The Golden

THE GOLDEN RATIO also known as The Golden Mean The Golden Section

Special Numbers

Pythagoras The square on the hypotenuse is equal to the sum of the squares on the other two sides 5 4 32 + 4 2 = 52 9 + 16 = 25 3

What is the Golden Ratio?

Curious Properties of The Golden Ratio

How to Draw a Golden Rectangle 2 1 1

The Fibonacci Series Each number is the sum of the 2 previous numbers

Fibonacci Convergence

A Logarithmic Spiral

The Star & The Golden Ratio 2

The Golden Triangle A How 36 E many golden triangles 108 D 72 can you see? 36 72 36 C B

The Golden Ratio and the 3 -4 -5 Triangle AB = 5 AC = 4 BC = 3 PQ =φ BP

The Golden Ratio and The Equilateral Triangle AB = BC = CA AL = BL AM = CM

The Golden Ratio and Circles

4 th Degree Polynomials

Bisecting the Yin-Yang Symbol

The Golden Ratio & The Semicircle When 3 equal circles touch each other in sequence, and a larger semicircle, the ratio of the radius of the semicircle to the diameter of the small circles is φ.

Areas of Annulus & Ellipse The area of an annulus defined by the radii, a and b (b < a) is equal to the area of an ellipse with major and minor axis radii equal to a and b, respectively, when a/b = φ b b a a

The Golden Ratio in Nature Two examples of flower heads containing multiple seeds in spiral arrays

The Golden Ratio & Seed Spirals 21 clockwise spirals 13 anticlockwise spirals

The Golden Ratio & Seed Spirals

Are these Golden Ratios?

Fact or Myth? Storms and galaxies are often spirals but are they Golden Ratio spirals?

The Golden Ratio & Trees

The Golden Ratio & Creatures

The Golden End