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Presentation on theme: "Fibonacci."— Presentation transcript:

1 Fibonacci

2 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….

3 I, II, III, IV, V, VI, VII, VIII, IX, X L, C, D, M
Arabic Numerals Roman Numerals I, II, III, IV, V, VI, VII, VIII, IX, X L, C, D, M Compare example: = MCMLIV L=50; C=100; D=500; M= Romans used a duodecimal system for fractions because of the divisibility of 12 (1, 2, 3, 4, 6, 12) Made it easier to handle common fractions ½, 1/3, ¼ Try operations * /

4 Leonardo Pisano Fibonacci born 1170 died 1250 in Pisa, Italy
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.

5 Leonardo Pisano Fibonacci
The problem….

6 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? F(n) = F(n-1) + F(n-2) for n = 3, 4, 5, ...

7 Fibonacci Patterns Around Us


9 Fibonacci Spiral

10 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

11 Chameleon Tail

12 Pinecone


14 Chambered Nautilus


16 Tornado March 2007 Silverton, TX



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




23 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,


25 137.5⁰

26 This angle is very important in describing how primordia form the spirals
to understand how these spirals come to be, one must go back to the beginning; to where flowers and fruits and seeds start: the apex. The apex is the tip of the shoot of a growing plant. It is the bud on the end of a stem on a tree and the bulb of a flower before it blooms. Around the apex grow little bumps called primordia. As more primordia develop, they are pushed farther and farther from the apex and they develop into the familiar features of a plant, be it a leaf, a flower, or parts of a fruit. Let us consider a sunflower with primordia growing from the center. The first primordia to develop end up being farther from the apex than later primordia. Therefore, it can be deduced from this in what order the primordia appeared. As it happens, if one took the first and second primordia and measured the angle between them with the center of the seed head as the vertex, the angle would be very close to degrees 137.5⁰

27 found in pineapple growth
Spirals found in pineapple growth seedhead


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

30 Aloe Phyllotaxis – pattern of leaves on a plant
Greek : phýllon "leaf" and táxis "arrangement”

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

32 1 Petal Calla Lily

33 2 Petals Euphorbia

34 3 Petals Trillium

35 5 Petals Columbine

36 8 Petals Bloodroot

37 13 Petals Brown Eyed Susan

38 21 Petals Shasta Daisy Daisies with 13, 21, 34, 55 or 89 petals are quite common.

39 341 Petals Field Daisy In saying that daisies have 34 petals, one is generalizing about the species - but any individual member of the species may deviate from this general pattern. There is more likelihood of a possible under development than over-development, so that 33 is more common than 35.





44 Pascal Triangle And Fibonacci



47 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

48 Mandalas Both ancient and modern Incorporate spiral patterns or radial symmetry

49 Tibetan Mandala

50 Buddhism Mandala

51 Hindu Mandala

52 Zulu Basket Phone wires

53 Sea Shells

54 Staircase Kew Garden London


56 Phi Greek Letter

57 = Golden Ratio Phi – math symbol 1.618033988749894848204586834…
Is irrational

58 Golden Ratio + 1 =

59 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 Ratio of successive Fibonacci numbers converge on phi




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

64 Vitruvian Man Proportional Ratio

65 Vitruvian Man Proportional Ratio


67 Human health is affected by facial proportions
Human health is affected by facial proportions. Biologically, people who have long faces have more chances of having breathing problem, suffering from sleep apnea. And People with shorter faces tend to have abnormal jaw development due to the excessive pressure on the jaw joint, suffering from headaches because their jaws are positioned in a manner that can restrict blood flow to the brain.


69 The Marquardt Beauty Mask
Dr. Stephen Marquardt has studied human beauty for years in his practice of oral and maxillofacial surgery. Dr. Marquardt performed cross-cultural surveys on beauty and found that all groups had the same perceptions of facial beauty. He also analyzed the human face from ancient times to the modern day. Through his research, he discovered that beauty is not only related to phi, but can be defined for both genders and for all races, cultures and eras with the beauty mask which he developed and patented. This mask uses the pentagon and decagon as its foundation, which embody phi in all their dimensions. The Marquardt Beauty Mask Asian Black Caucasian

70 164 A.D. Rome 1794 A.D. 1350 B.C. Egypt 500 B.C. Greece


72 Golden Rectangle

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

74 Leonardo DaVinci Used the Golden Ratio Mona Lisa

75 Composition in Red, Yellow, and Blue
Piet Mondrian 1926

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

77 Parthenon Parthenon

78 Tajamahall

79 Approximate and true golden spirals
Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of one square divided by that of the next smaller square is the golden ratio


81 Musical frequencies based on Fibonacci ratios
Notes in the scale of western music are based on natural harmonics that are created by ratios of frequencies. Ratios found in the first seven numbers of the Fibonacci series ( 0, 1, 1, 2, 3, 5, 8 ) are related to key frequencies of musical notes. Musical frequencies are based on Fibonacci ratios

82 Musical instrument design often based on phi
the golden ratio

83 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

84 Professor of Mechanical Engineering at Duke University
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 Bejan argues that an animal's world – whether you are a human being in an art gallery or an antelope on the savannah – is orientated on the horizontal. For the antelope scanning the horizon, danger primarily comes from the sides or from behind, not from below or above, so the scope of its vision evolved accordingly. As vision developed, he argues, animals got "smarter" and safer by seeing better and moving faster as a result.

85 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."

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

87 The Fibonacci Sequence and the Golden Ratio to Music

88 The Fibonacci Sequence
Discussion BBC


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