maths and nature for layman
TRANSCRIPT
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Patterns, Nature, & Math
Are They Interconnected?
Mrs. Christi Wilson
Greenbrier Middle School
Maths of nature
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•“If people do not believe
that mathematics is simple,it is only because they do
not realize how complicated life is “
(John Von Neumann)
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Tree Trunk Mushroom
Plants
Granite Fossils
Minerals
Butterfly Tiger
Animals
Waves Dew
Water
"Natural" Patterns
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Natural beauties
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Natural Intelligence
• Some Insects can performs
complicated Tasks which is
beyond Human beings.
• Unknowingly they apply quite
advance mathematics.
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spider’s web
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spider’s web
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spider’s web
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spider’s web
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spider’s web
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Spider’s web is 25 times stronger than
the same thickness steel wire
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• Termites air-conditioned
mound with chimney.
• It is an EngineeringMarvel.
•
Up to 30 feet high.• CO2 /O2 balance .
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S
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Square
pattern
Non Useful
empty space
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Non Useful
empty space
Hexagonal pattern
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Hexagonal packing
Fibonacci Numbers
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Fibonacci Numbers
Nature has it’s own Number
System called Fibonacci
number system which is
applicable to all those
things which take birthand grow primarily and
almost all aspects of life
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Fibonacci “ Leonardo Da vinci”
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Role of Fibonacci numbers
• Trees , Plants, leaves, flowers, stems
• Bee Hive, spider’s web • Human Body
• Various insects an Animals• music, art, Culture, and Science
• Business, Ads, and share market
St t ith i f bbit
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Start with one pair of rabbits
Rules of growth
1. Every new born pair becomes adult in
one month.
2. Every adult pair after one month givesbirth to one pair every month.
3. No rabbit dies.
population of rabbits after one year = ?
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Modelization of a population
• First month
• Secod month
• Sixth month
• third month
• Fourth month
• Fifth month
Adult pairs Young pairs
Sequence: 1, 1, 2, 3, 5, 8, … Fn+1= Fn + Fn-1
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Month Pairs of Rabbits
1 1
2 1
3 2
4 3
5 5
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Fibonacci Series
• The rule for the sequence?
1, 1, 2, 3, ?, ?, ….
• The Answer :1, 1, 2, 3, 5, 8, 13, 21, …
1+ 1 = 21+ 2 = 32+ 3 = 53+ 5 = 85+ 8 = 138+ 13 = 21
Fibonacci’s Rabbit Colony
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Fibonacci s Rabbit Colony
1 1 2 3 5 8 13 2134
55
89
144
0
20
40
60
80
100
120
140
160
J a n u a r y
F e b r u a r y
M a r c h
A p r i l
M a y
J u n e
J u l y
A u g u s t
S e p t e m b e r
O c t o b e r
N o v e m b e r
D e c e m b e r
Pairs of Rabbits
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New born tender
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‘sneezewort’
One
One
Two
Three
Five
Eight
New born ---- tenderAdult ---- strongGives birth ----branches out
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1 1 2 3 5 8 13 21 34 55
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1, 1 ,2 ,3 ,5 ,8 ,13 ,21 ,34 ,55 ,…
1/1, 2/1, 3/2, 5/3, 8/5, 13/8…
1, 2, 1.5, 1.666, 1.6, 1.625, …
Golden ratio
phi = 1.6180339887…
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Phi = 1 6180339887
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Phi = 1.6180339887…
• Golden ratio
• Golden mean
• Golden section
• Fibonacci ratio
• Divine Ratio
• Sacred ratio
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<>
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Golden Spiral manipulative
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(Sh ll)
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(Shell)
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The human feotus develops along the lines of a Fibonacci spiral
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Sheep
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Sheep
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Galaxies
Sea Shell
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A low pressure area over Iceland showsan approximately logarithmic spiral pattern
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Spirals in the Galaxy
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p y
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Rose petals
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Sunflower with 34 petals
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Sunflower with 34 petals
1 13
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Flowers!1
2
3
5
8
13
21
white calla lilyeuphorbia trillium
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white calla lily
bloodroot black-eyed susan
daisy family
13, 21, 34, 55
or 89 petals
Leaf arrangements
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g
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Leaves, petals and
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, p
seeds arranegment
0·382x360 = 137·5°
222.5/137.5 =Φ
= 1.618
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• In a 120 0 arrangement like a fan,
4th one would hide the first leaf
Leaf arrangements
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g• If we look down on a plant, the leaves are
often arranged so that leaves above do nothide leaves below.
• Each gets a good share of the sunlight and
catches the most rain to channel down to theroots as it runs down the leaf to the stem.
• Insects can collect pollens easily.
Why does nature like using Phi in so many plants?
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• Although hexagonal symmetry is
the best packing for circular seeds,
it doesn't answer the question of
how leaves should be arranged
round a stem or how to pack
flower-heads with seeds that grow in size.
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Fi di G ld i
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Finding Golden ratio
• As seeds grow older they are pushedout.
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Left to right: A sunflower picture from The Algorithmic Beauty of Plants , P.
Prusinkiewicz. A cactus by C. Smith. Another Cactus by C. Smith and M.Fuhrer.
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Pine cones pa[na kaona
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Banana Apple
In vegetable?
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5
Red
Spirals
8
Blue
Spirals
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1 1.618
WHOLE
WHOLE = LARGE = GOLDEN MEAN
LARGE SMALL
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Human Hand
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Human Hand
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DNA
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DNA
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• Dr. StephenMarquardt, a former
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q ,plastic surgeon, hasused the goldensection, thatenigmatic numberthat has long stood for
beauty, and some of its relatives to make amask that he claims isthe most beautiful
shape a human facecan have.
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Leonardo Da Vinci, Mona
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स धम समभा
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Symbol
Musical instruments are often based
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on phi
Musical scales are based on
Fibonacci numbers;
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Fibonacci numbers;
piano keyboard scale of 13
keys has 8 white keys and 5black keys, split into groups of
3 and 2.
• 13 notes separate each octave of 8 notes in a scale,
of which the 5th and 3rd notes create the basic
foundation of all chords, and are based on whole
tone which is 2 steps from the root tone, that is
the 1st note of the scale.
Golden mean gauge
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gaaolDna Anaupat maapI
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The Bagdad City Gate
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Dome ofSt. Paul:London,England
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Windson Castle
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Golden Caliper
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Advertisement
ART
rchitecture
MUSIC
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Human expectations occurin a ratio that approaches Phi
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•
our brains are hardwired to find Fibonacciprogressions naturally pleasing.
• Changes in stock prices largely reflect human
opinions, valuations andexpectations. Humans exhibit positive and
negative evaluations of the opinions they hold
in a ratio that approaches phi, with 61.8%positive and 38.2% negative.
• Used to predicting stock market behaviour.
Shape of Universe
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Rings of Saturn
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Further classic subdivisions of the rectanglealign perfectly with major architectural features
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Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure.
align perfectly with major architectural featuresof the structure.
Egyptians in the design of the pyramids,
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•The CN Tower in Toronto,the tallest tower.
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•The ratio of observationdeck at 342 meters to thetotal height of 553.33 is0.618, the reciprocal of Phi!
Notre Dame in Paris
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2
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8 5
2
1
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Phi as repeated square root
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Phi as continued fraction
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Fibonacci numbers define key points in human aging
Huma
n Age
Development Stage
Key Attributes
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0 Gestation Conception
1 Newborn Birth
1 Infant Walking, vocalizing
2 Toddler Talking, expressing, imitating
3 Toddler Self image and control, toilet training
5 Early child Formal education begins
8 Mid child Age of reason, knowing of right and wrong
13 Adolescent Thinking, puberty, sexual maturation and drive
21 Youngadult
Full physical growth, adult in society, educationcomplete, beginning career, financial
responsibility, eligible for voting34 Mid adult Refinement of adult skills, parenting role
55 Elder adult Fulfillment of adult skills, serving, retirement beginswith eligibility for Medicare, Social Security
89 Completion Insight and wisdom into life
Surely it is logical to conclude
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that the connections exist
because nature, mathematicsand the human mind, with its
subtle sense of beauty, have onesupreme link— they are all the
created products of God, theMaster Designer - Russel Grigg