Transcript Tornado March 2007 Silverton, TX

Fibonacci
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….
Arabic Numerals
Roman Numerals
I, II, III, IV, V, VI, VII, VIII, IX, X
L, C, D, M
Compare example: 1954 = MCMLIV
Leonardo Pisano Fibonacci
born 1170
died 1250 in Pisa, Italy
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
The problem….
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•
•
•
•
•
To start, there are no bunniers.
The first month, 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?
Fibonacci
Patterns Around Us
Fibonacci Spiral
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
Chameleon Tail
Pinecone
Chambered
Nautilus
Tornado
March 2007
Silverton, TX
Fibonacci
Found in
• Sunflowers - seeds
• Cauliflowers - florets
Nature’s efficiency problem
• Fibonacci Sequence finds the most efficient
packing system
137.5⁰
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,
1920-1961
This angle is
very
important in
describing
how
primordia
form the
spirals
137.5⁰
Spirals
found in
pineapple
growth
Fibonacci
Found in
• Leaves on a stem – sunshine energy reaches
leave most efficiently
Nature’s problem
• Fibonacci Sequence finds the most efficient
distribution system
Aloe
Cut open a red pepper, lemon, or apple. How
many chambers, sections, or seeds are there?
1 Petal
Calla Lily
2 Petals
Euphorbia
3 Petals
Trillium
5 Petals
Columbine
8 Petals
Bloodroot
13 Petals
Brown Eyed Susan
21 Petals
Shasta Daisy
341 Petals
Field Daisy
Pascal Triangle
And
Fibonacci
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
Mandalas
Both ancient and modern
Incorporate
spiral patterns or
radial symmetry
Tibetan
Mandala
Buddhism
Mandala
Hindu
Mandala
Zulu
Basket
Sea
Shells
Staircase
Kew Garden
London
Phi
Greek Letter
Golden Ratio
Phi – math symbol

=
1 5

2
1.618033988749894848204586834…
Is irrational
Golden Ratio
 + 1 =
2
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
Assume a+b = 1
then b = 1-a
1
𝑎
=
𝑎
1−𝑎
Then a2 = 1 – a
a2 +a -1 = 0
Rewrite a = x.
−𝑏 ± 𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
Then
x=
x=
x=
−1± 12 −4(1)(−1)
2(1)
5+1
= 1.618…
2
5−1
= 0.618…
2
Golden Ratio
Fibonacci Sequence is a growth pattern
Found in nature. Man uses it to design.
Designed for is aesthetically pleasing ratio

1.
2.
3.
4.
5.
Postcard
Credit Card
Parthenon
Car
Human Body smile, face…
face
Aston Martin
James Bond
and the Golden Ratio
The ‘Golden Ratio’ sits at the heart of every
Aston Martin. Balanced from any angle,
each exterior line.. works in concert and
every proportion is precisely measured to
create a lithe, pure form.
Aston Martin Engineer
Parthenon
Parthenon
Vitruvian Man
Proportional
Ratio
Vitruvian Man
Proportional
Ratio
The Marquardt Beauty Mask
Asian
Black
Caucasian
1350 B.C.
Egypt
164 A.D. Rome
500 B.C.
Greece
1794 A.D.
Golden Rectangle
Salvador Dali
Used the Golden Ratio
The Sacrament of the Last Supper
Leonardo DaVinci
Used the Golden Ratio
Mona Lisa
Composition in Red, Yellow, and Blue
Piet Mondrian
1926
Pentagon
a/b
=(a+b)/a
=(a+b+a)/(a+b)
=phi
=Golden Ratio
Parthenon
Taj Mahal
Eiffel Tower
Musical frequencies
based on
Fibonacci ratios
Musical frequencies are based on Fibonacci ratios
Musical instrument design
often based on phi
the golden ratio
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
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 pattern.
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."
Adrian Bejan
Professor of Mechanical Engineering at Duke
University
Durham, North Carolina
“This ratio represents the best proportions to transfer
to the brain.”
The Fibonacci Sequence and the Golden Ratio to Music
The Fibonacci Sequence
Discussion
BBC
Blaise Pascal
1623-1662
French Mathematician
Pascal - computer
language is named
after him. When 19,
he invented a
calculating machine ~
the early form of
today’s computers.
Worked on probability
with Fermat.
Pascal Triangle
One of the best known integer patterns in mathematics.
Pascal Triangle
Pascal Triangle
Gaussian Distribution - a function that tells the probability
that any real observation will fall
between any two real limits as the
curve approaches zero on either
side
Hockey Stick Pattern
1+7+28+84+210+462+924 = 1716
1+6+21+56 = 84
1+12 = 13
Pascal Triangle
Taxi Cab Geometry
If you can only
move in the
directions of the
arrows (right and
down, not up or
left), how many
paths are there
from A to B?
Sierpinski Triangle
If color in only odd numbers from the
Pascal Triangle, you get a fractal Sierpinski Triangle