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Fibonacci

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Fibonacci of Pisa Italian Middle Ages Mathematician Brought Hindu Arabic Numeral System to Europe Through the publication in the early 13th century of his Book of Calculation, the Liber Abaci. Posed a problem in the book….

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Arabic Numerals I, II, III, IV, V, VI, VII, VIII, IX, X L, C, D, M Compare example: 1954 = MCMLIV Roman Numerals

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Leonardo Pisano Fibonacci – Fibonacci - son of the Fibonaccis… Pisano – from where he lived most of his life, Pisa Pisa, Italy - famous for its leaning tower Building of the tower began when Fibonacci was 3 years old. Leonardo Pisano Fibonacci born 1170 died 1250 in Pisa, Italy

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The problem…. Leonardo Pisano Fibonacci

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A man puts a pair of baby rabbits into an enclosed garden. The following month the rabbits are teenagers. The next month they are mature and have two babies. Continue this pattern. How many pairs of rabbits will there be in the garden in a year?

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Fibonacci Patterns Around Us

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Fibonacci Spiral

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Fibonacci Sequence Demonstrates the idea that each stage of natural growth or development refers to its prior state in order to take its next evolutionary step This connects to the idea of Sacred Geometry

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Chameleon Tail

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Pinecone

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Chambered Nautilus

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Tornado March 2007 Silverton, TX

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Fibonacci Found in Sunflowers - seeds Cauliflowers - florets Nature’s packing problem Fibonacci Sequence finds the most efficient packing system

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Le Tournesol may be Steichen's only sunflower painting, but his fascination with this plant is evident in the numerous photographic studies he made in the 1920s. He photographed Sunflower in a White Vase at this time. Almost twenty years earlier, Steichen had been impressed with other sunflowers: the painted still lifes of Vincent van Gogh. In 1901, he visited an exhibition of Van Gogh's work, which left a deep impression on him. He returned to the exhibition the next day, noting later that "three pictures of the now celebrated sunflower series made a particular and dramatic appeal to me." Edward Steichen, Sunflower in a White Vase, from the series Sunflowers from Seed to Seed,

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137.5⁰

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This angle is very important in describing how primordia form the spirals 137.5⁰

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Spirals found in pineapple growth

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Fibonacci Found in Leaves on a stem – sunshine energy reaches leave most efficiently Nature’s problem Fibonacci Sequence finds the most efficient distribution system

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Aloe

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Cut open a red pepper, lemon, or apple. How many chambers, sections, or seeds are there?

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1 Petal Calla Lily

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2 Petals Euphorbia

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3 Petals Trillium

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5 Petals Columbine

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8 Petals Bloodroot

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13 Petals Brown Eyed Susan

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21 Petals Shasta Daisy

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341 Petals Field Daisy

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Pascal Triangle And Fibonacci

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Mandalas Meaning circle Sanskirt word. For many religious traditions, circles are found in sacred art In the Buddhist and Hindu religious traditions their sacred art often takes a Mandala form

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Mandalas Both ancient and modern Incorporate spiral patterns or radial symmetry

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Tibetan Mandala

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Buddhism Mandala

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Hindu Mandala

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Zulu Basket

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Sea Shells

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Staircase Kew Garden London

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Phi Greek Letter

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Golden Ratio Phi – math symbol = … Is irrational

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Golden Ratio + 1 =

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Ratio of successive Fibonacci numbers converge on phi Fibonacci discovered the series which converges on phi The ratio of each successive pair of numbers in the series approximates phi Example 5 divided by 3 is 1.666… 8 divided by 5 is 1.60

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Golden Ratio 1.Postcard 2.Credit Card 3.Parthenon 4.Human Body smile, face… face Fibonacci Sequence is a growth pattern Found in nature Designed for is aesthetically pleasing ratio

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Vitruvian Man Proportional Ratio

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Vitruvian Man Proportional Ratio

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The Marquardt Beauty Mask AsianBlackCaucasian

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1350 B.C. Egypt 500 B.C. Greece 164 A.D. Rome1794 A.D.

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Golden Rectangle

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Salvador Dali Used the Golden Ratio The Sacrament of the Last Supper

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Leonardo DaVinci Used the Golden Ratio Mona Lisa

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Composition in Red, Yellow, and Blue Piet Mondrian 1926

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a/b =(a+b)/a =(a+b+a)/(a+b) =phi =Golden Ratio Pentagon

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Parthenon

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Tajamahall

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Musical frequencies based on Fibonacci ratios Musical frequencies are based on Fibonacci ratios

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Musical instrument design often based on phi the golden ratio

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Musical instrument: piano Within the scale consisting of 13 keys 8 of them are white, 5 are black which are split into groups of 3 and 2 Fibonacci Music

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Adrian Bejan Professor of Mechanical Engineering at Duke University Durham, North Carolina Research The human eye is capable of interpreting an image featuring the golden ratio faster than any other

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Adrian Bejan Professor of Mechanical Engineering at Duke University Durham, North Carolina "It is well known that the eyes take in information more efficiently when they scan side to side, as opposed to up and down. When you look at what so many people have been drawing and building, you see these proportions everywhere."

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Adrian Bejan Professor of Mechanical Engineering at Duke University Durham, North Carolina “This ratio represents the best proportions to transfer to the brain.”

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The Fibonacci Sequence and the Golden Ratio to Music

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The Fibonacci Sequence Discussion BBC

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