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Golden Ratio Phi Φ

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1. Golden Ratio Phi Φ 1 : 1.618 * Fibonacci’s series 1 1 2 3 5 8 13 21 1 1 2 3 5 8 13 21 1/1 = 13/2 = 1.58/5 = 1.621/13 = 1.615… 1/1 = 13/2 = 1.58/5 = 1.621/13 = 1.615… 2/1 = 2 5/3 = 1.666... 13/8 = 1.625 2/1 = 2 5/3 = 1.666... 13/8 = 1.625

2. Golden Ratio in the Human Body Click me! Interactive Golden Ratio Mask

3. Golden Spiral • If it grows outward by a factor of the golden ratio for every 90 degrees of rotation known as a ‘golden spiral’ • One approximation of the ‘golden spiral’ can be formed using Fibonacci Rectangles

4. Fibonnaci Rectangles Approximating the Golden Spiral

5. Logarithmic spirals • Sea shells follow a spiral that widens as it winds around itself • Discovered by Rene Descartes in 1638 • He found the lines drawn from the center of the spiral intersected with tangents to the spiral at a constant angle • Also called an equiangular spiral

6. Maths behind Logarithmic Spiral… • The logarithmic spiral is a spiral whose polar equation is given by • e is the base of natural logarithms, and a and b are positive real constants • ‘a’ controls the size of the spiral • ‘b’ controls how ‘tightly’ the spiral winds • As it tends to 0, the spiral tends towards a circle of radius ‘a’ • As it tends to ∞ , the spiral tends towards a straight line