The Golden Mean

1. The Golden Mean The Mathematical Formula of Life

2. The Golden Mean • The Golden Mean is a ratio which has fascinated generation after generation, and culture after culture. It can be expressed succinctly in the ratio of the number "1" to the irrational “l.618034.”

3. 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

4. The Golden Mean • The golden ratio is 1·618034. It is often represented by a Greek letter Phi Φ. • The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... (add the last two to get the next) • The golden ratio and Fibonacci numbers relate in such that sea shell shapes, branching plants, flower petals and seeds, leaves and petal arrangements, all involve the Fibonacci numbers.

5. A M B One Way to Understand It • The line AB is divided at point M so that the ratio of the two parts, the smaller MB to the larger AM is the same as the ratio of the larger part AM to the whole AB. Does that make sense?

6. OR Given a rectangle having sides in the ratio 1:phi , phi is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio 1: phi. Such a rectangle is called a golden rectangle, and successive points dividing a golden rectangle into squares lie on a logarithmic spiral. This figure is known as a whirling square.

7. Have You Seen This? Note that each new square has a side which is as long as the sum of the latest two square's sides.

8. The Golden Mean and Aesthetics • Throughout history, the ratio for length to width of rectangles of 1.61803 39887 49894 84820 has been considered the most pleasing to the eye. • Artists use the Golden Mean in the creation of great works.

9. The Parthenon • “Phi“ was named for the Greek sculptor Phidias. • The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.

10. Leonardo Da Vinci • Many artists who lived after Phidias have used this proportion. Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings, for example in the famous "Mona Lisa". Try drawing a rectangle around her face. Are the measurements in a golden proportion? You can further explore this by subdividing the rectangle formed by using her eyes as a horizontal divider.

11. The “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.

12. Look at your own hand: • You have ... • 2 hands each of which has ... • 5 fingers, each of which has ... • 3 parts separated by ... • 2 knuckles • Is this just a coincidence or not?????

13. The Golden Mean is Also Found in Nature

14. The Golden Spiral can be seen in the arrangement of seeds on flower heads.

15. Pine cones show the Fibonacci Spirals clearly. Here is a picture of an ordinary pinecone seen from its base where the stalk connects it to the tree.

16. On many plants, the number of petals 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 whereas daisies can be found with 34, 55 or even 89 petals.

17. Patterns of Nature • In this assignment students will choose a pattern from nature which is created through the phenomenon of the Golden Mean such as the pattern in a Nautilus Shell and create an original design. • Students will use this pattern, or the one demonstrated in the “Last Supper” to create an original work of art using the Golden Mean to create the composition. • The solutions to this problem are infinite.

18. Ideas for Designs • The pattern of a butterfly wing…. • Patterns of sea creatures…… • Close ups of patterns from nature….. • Leaf arrangements, leaf veins, petal patterns…. • Feather patterns from birds such as one feather, or the entire tail pattern of a peacock…. • Look at animals, bugs, fish, and plants to get ideas…..

19. Rubric • The design must be original. • The composition must use the Golden Mean. • The painting must use a pattern found in nature to inspire the design. • The design must demonstrate knowledge of space as an element of design. • The project is to be done in acrylic paint. • The design must show technical craftsmanship. • The student must use proper care and conservation of tools and supplies.

20. The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and science. He who knows it not and can no longer wonder, no longer feels amazement, is as good as dead, a snuffed-out candle. —Albert Einstein