The Golden Mean The Mathema)cal Formula… of Life?
The Golden Mean Also known as:• The Golden Ratio• The Golden Section• The Golden Rectangle• The Golden Number• The Golden Spiral• Or the Divine Proportion
The Golden Mean The Golden Mean is a ratio whichhas fascinated generation aftergeneration, and culture after culture.
The Golden Mean It can be expressed succinctly in theratio of the number “1” to theirrational 1.618034. 1.618034 1
The Golden Mean • The golden ratio is 1·618034. It is often represented by a Greek letter Phi Φ. Phi pronuncia)on: Linguis)c purist might opt for the original Greek fee, most mathema)cians know phi as ﬁ. • The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... (add the last two to get the next and so on)• The golden ratio and Fibonacci numbers also exist in nature… more on that in a bit.
One Way to Understand It Is A M B A B A MThe line AB is divided at point M so thatthe ratio of the two parts,the smaller MB to the larger AM is thesame as the ratio of the larger part AM tothe whole AB. Does that make sense?
So the Golden Ratio is… Given a rectangle having sides inthe ratio 1:phi, phi is defined suchthat partitioning the originalrectangle into a square and newrectangle results in a newrectangle having sides with a ratio1: phi. Such a rectangle is calleda golden rectangle, andsuccessive points dividing agolden rectangle into squares lieon a logarithmic spiral. This figureis known as a whirling square.
The Golden Mean & Aesthe1cs • Throughout history, the ratio for length to width of rectangles of 1: 1.61803 39887 49894 84820 has been considered the most pleasing to the eye
The Parthenon The exteriordimensions ofthe Parthenon inAthens
The Parthenon The exteriordimensions ofthe Parthenon inAthens, built inabout 440BC …
The Parthenon The exteriordimensions ofthe Parthenon inAthens, built inabout 440BC …form a perfectgolden rectangle.
Golden SpiralLogarithmic Spiral: The Whirling Square.Note that each new square has a side which is as long as the sum ofthe latest two squares sides.
Leonardo Da Vinci • Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings.
Leonardo Da Vinci • Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings.• In the famous "Mona Lisa". Try drawing a rectangle around her face. You can further explore this by subdividing the rectangle formed by using her eyes as a horizontal divider, etc.
Vitruvian Man • Leonardo did an entire exploration of the human body and the ratios of the lengths of various body parts. Vitruvian Man illustrates that the human body is proportioned according to the Golden Ratio.
Look at your own hand:You have ...• 2 hands each of which has ...• 5 fingers, each of which has ...• 3 parts separated by ...• 2 knucklesAnd…
On many plants, the number ofpetals is a Fibonacci number:buttercups have 5 petals;lilies and iris have 3 petals;some delphiniums have 8;corn marigolds have 13 petals;some asters have 21 whereasdaisies can be found with 34,55 or even 89 petals.
The Golden Spiral can be seen in thearrangement of seeds on flower heads.
Pine cones showthe FibonacciSpirals clearly.Here is a pictureof an ordinarypinecone seenfrom its basewhere the stalkconnects it tothe tree.
Project 8 • Objectives We will explore geometric relationships within the Picture plane derived from the Golden Section, or the Golden Rectangle. This system provides a way to place and organize shapes based on using mathematical ratios and rational proportions. We will also explore how to direct the viewer s eye by contrasts of value, shape and size.