Golden Ratio Phi Φ

2/1 = 2 5/3 = 1.666... 13/8 = 1.625

1 1 2 3 5 8 13 21 1/1 = 1 3/2 = 1.5 8/5 = 1.6 21/13 = 1.615…

1 : 1.618

2/1 = 2 5/3 = 1.666... 13/8 = 1.625

1 1 2 3 5 8 13 21 1/1 = 1 3/2 = 1.5 8/5 = 1.6 21/13 = 1.615…

* Fibonacci’s series

Golden Ratio in the Human Body

Interactive Golden Ratio MaskClick me!

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

Fibonnaci Rectangles Approximating the Golden Spiral

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

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