The Golden MeanThe Golden Mean

The Mathematical Formula ofThe Mathematical Formula of

LifeLife

The Golden Mean The Golden Mean

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

The Golden MeanThe Golden Mean

Also known as: Also known as: The Golden RatioThe Golden Ratio The Golden SectionThe Golden Section The Golden RectangleThe Golden Rectangle The Golden NumberThe Golden Number The Golden SpiralThe Golden Spiral Or the Divine ProportionOr the Divine Proportion

The Golden Mean The Golden Mean

The The golden ratiogolden ratio is 1·618034. It is often is 1·618034. It is often represented by a Greek letter represented by a Greek letter Phi Phi Φ.Φ.

The The Fibonacci numbersFibonacci numbers are are 0, 1, 1, 2, 3, 5, 8, 0, 1, 1, 2, 3, 5, 8, 13, ... 13, ... (add the last two to get the next)(add the last two to get the next)

The golden ratio and Fibonacci numbers relate The golden ratio and Fibonacci numbers relate in such that sea in such that sea shellshell shapes, branching plants, shapes, branching plants, flower petals and seeds, leaves and petal flower petals and seeds, leaves and petal arrangements, all involve the Fibonacci arrangements, all involve the Fibonacci numbers. numbers.

A M BA M B

The line AB is divided at point M so that the The line AB is divided at point M so that the ratio of the two parts, the smaller MB to the ratio of the two parts, the smaller MB to the larger AM is the same as the ratio of the larger AM is the same as the ratio of the larger part AM to the whole AB. Does that larger part AM to the whole AB. Does that make sense?make sense?

One Way to Understand It

ORORGiven 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.

Have You Seen This?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.

The Golden Mean and AestheticsThe Golden Mean and Aesthetics

Throughout history, the ratio for length to Throughout history, the ratio for length to width of rectangles of 1.61803 39887 49894 width of rectangles of 1.61803 39887 49894 84820 has been considered the most 84820 has been considered the most pleasing to the eye. pleasing to the eye.

Artists use the Golden Mean in the creation Artists use the Golden Mean in the creation of great works.of great works.

The ParthenonThe Parthenon

““Phi“ was named Phi“ was named for the Greek for the Greek sculptor Phidias. sculptor Phidias.

The exterior The exterior dimensions of the dimensions of the Parthenon in Parthenon in Athens, built in Athens, built in about 440BC, about 440BC, form a perfect form a perfect golden rectangle.golden rectangle.

Leonardo Da VinciLeonardo Da Vinci Many artists who lived after Many artists who lived after

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

The “Vitruvian The “Vitruvian Man”Man”

Leonardo did an entire Leonardo did an entire exploration of the exploration of the human body and the human body and the ratios of the lengths of ratios of the lengths of various body parts. various body parts. “Vitruvian Man” “Vitruvian Man” illustrates that the illustrates that the human body is human body is proportioned according proportioned according to the Golden Ratio. 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 knuckles Is this just a coincidence or not?????

The Golden The Golden Mean is Mean is

Also Found Also Found in Naturein Nature

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

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.

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.

Patterns of Nature Patterns of Nature

In this assignment students will choose a pattern from In this assignment students will choose a pattern from nature which is created through the phenomenon of nature which is created through the phenomenon of the Golden Mean such as the pattern in a Nautilus the Golden Mean such as the pattern in a Nautilus Shell and create an original design. Shell and create an original design.

Students will use this pattern, or the one Students will use this pattern, or the one demonstrated in the “Last Supper” to create an demonstrated in the “Last Supper” to create an original work of art using the Golden Mean to create original work of art using the Golden Mean to create the composition.the composition.

The solutions to this problem are infinite. The solutions to this problem are infinite.

Ideas for DesignsIdeas for Designs

The pattern of a butterfly wing….The pattern of a butterfly wing…. Patterns of sea creatures……Patterns of sea creatures…… Close ups of patterns from nature…..Close ups of patterns from nature….. Leaf arrangements, leaf veins, petal patterns….Leaf arrangements, leaf veins, petal patterns…. Feather patterns from birds such as one feather, Feather patterns from birds such as one feather,

or the entire tail pattern of a peacock….or the entire tail pattern of a peacock…. Look at animals, bugs, fish, and plants to get Look at animals, bugs, fish, and plants to get

ideas…..ideas…..

RubricRubric The design must be original.The design must be original. The composition must use the Golden Mean.The composition must use the Golden Mean. The painting must use a pattern found in nature to The painting must use a pattern found in nature to

inspire the design.inspire the design. The design must demonstrate knowledge of space as The design must demonstrate knowledge of space as

an element of design.an element of design. The project is to be done in acrylic paint.The project is to be done in acrylic paint. The design must show technical craftsmanship.The design must show technical craftsmanship. The student must use proper care and conservation of The student must use proper care and conservation of

tools and supplies. tools and supplies.

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