Golden Ratio Phi Φ 1 : 1.618 This could be a fun lesson. Fibonacci was inspired by how fast rabbits could breed in ideal circumstances, counting how many pairs each breeding cycle produced, giving birth to a pair. Many things found in nature have parts that are in a ratio of 1.618:1. The Greeks said that all beauty is mathematics. http://www.intmath.com/Numbers/mathOfBeauty.php In Fibonacci numbers, this ratio is fixed after the thirteenth in series. www.allahexists.com/earth/earth_05.php The first example of the golden ratio in the average human body is that when the distance between the navel and the foot is taken as 1 unit, the height of a human being is equivalent to 1.618. Some other golden proportions in the average human body are: The distance between the finger tip and the elbow / distance between the wrist and the elbow, The distance between the shoulder line and the top of the head / head length, The distance between the navel and the top of the head / the distance between the shoulder line and the top of the head, http://www.highlandwoodworking.com/woodnews/july_2006/ask_the_staff_july06.html * Fibonacci’s series 1 1 2 3 5 8 13 21 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 = 1 3/2 = 1.5 8/5 = 1.6 21/13 = 1.615… 2/1 = 2 5/3 = 1.666... 13/8 = 1.625 2/1 = 2 5/3 = 1.666... 13/8 = 1.625

Golden Ratio in the Human Body While I am ‘doing’ photo, emma passes around pineapple etc. Can you guess what they have in common? Spirals! Length of face / width of face, Distance between the lips and where the eyebrows meet / length of nose, Length of face / distance between tip of jaw and where the eyebrows meet, Length of mouth / width of nose, Width of nose / distance between nostrils, Distance between pupils / distance between eyebrows. http://www.artofraz.com/freearttutorials/golden-ratio-art-tutorial.php http://www.highlandwoodworking.com/woodnews/july_2006/ask_the_staff_july06.html www.cogs.susx.ac.uk/courses/gc/lec14.html Click me! Interactive Golden Ratio Mask

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 This demonstrates how you can approximate the golden spiral using fibonnaci rectangles. This involves drawing 2 squares one above the other, then a larger square with side length of the first two combined. Then the next square will have a side length of the previous 2 combined and so on. This creates the golden ratio repeated throughout the overall picture, and connected the square corners creates an approximate 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