Post on 14-Jul-2015
Patterns, Nature, & Math Are They Interconnected?
Mrs. Christi Wilson
Greenbrier Middle School
Maths of nature
अमरनाथ मूर्ती
•“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is “
(John Von Neumann)
Tree Trunk Mushroom
Plants
Granite Fossils
Minerals
Butterfly Tiger
Animals
Waves Dew
Water
"Natural" Patterns
Natural beauties
Natural Intelligence
• Some Insects can performs complicated Tasks which is beyond Human beings.
• Unknowingly they apply quite advance mathematics.
spider’s web
spider’s web
spider’s web
spider’s web
spider’s web
Spider’s web is 25 times stronger than the same thickness steel wire
• Termites air-conditioned mound with chimney.
• It is an Engineering Marvel.
• Up to 30 feet high.
• CO2 /O2 balance .
• Inside these mounds are “gardens” where termites grow their food called fungus.
Square pattern
Non Useful empty space
Non Useful empty space
Hexagonal pattern
3 sides 4 sides 5 sides 6 sides
leaves gaps
The only regular polygons that can cover the plane are triangles, squares and hexagons.
Efficiency
Equillateral
Triangle Area
0.048
Area of Square
0.063
Area of hexagon
0.075
Packing Spheres
Hexagonal packing
Fibonacci Numbers
Nature has it’s own Number System called Fibonacci number system which is applicable to all those things which take birth and grow or grow otherwise ; almost all aspects of life.
Fibonacci “ Leonardo Da vinci”
Role of Fibonacci numbers • Trees , Plants, leaves, flowers, stems
• Bee Hive, spider’s web
• Human Body
• Various insects and Animals
• music, art, Culture, and Science
• Business, Ads, and share market
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 gives birth to one pair every month.
3. No rabbit dies.
population of rabbits after one year = ?
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, …
Month Pairs of Rabbits
1 1
2 1
3 2
4 3
5 5
Fibonacci Series
• The rule for the sequence?
1, 1, 2, 3, ?, ?, ….
• The Answer : 1, 1, 2, 3, 5, 8, 13, 21, …
1+ 1 = 2 1+ 2 = 3 2+ 3 = 5 3+ 5 = 8 5+ 8 = 13 8+ 13 = 21
Fn+1= Fn + Fn-1
Fibonacci’s Rabbit Colony
1 1 2 3 5 8 13 2134
55
89
144
0
20
40
60
80
100
120
140
160
Janu
ary
Feb
rua
ry
Ma
rch
Ap
ril
Ma
y
June
July
Au
gu
st
Sep
tem
ber
Oct
ob
er
Nov
em
ber
Dece
mb
er
Pairs of Rabbits
‘sneezewort’
One
One
Two
Three
Five
Eight
New born ---- tender Adult ---- strong Gives birth ----branches out
<>
Golden Spiral manipulative
(Shell)
The human feotus develops along the lines of a Fibonacci spiral
Sheep
Galaxies
Sea Shell
A low pressure area over Iceland shows an approximately logarithmic spiral pattern
Spirals in the Galaxy
Rose petals
Sunflower with 34 petals
Flowers! 1
2
3
5
8
13
21
white calla lily euphorbia trillium
bloodroot black-eyed susan
daisy family
13, 21, 34, 55
or 89 petals
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…
Phi = 1.6180339887…
• Golden ratio
• Golden mean
• Golden section
• Fibonacci ratio
• Divine Ratio
• Sacred ratio
Leaf arrangements
1 3 6
8 9
10
11 12
13 1
4
15
2 5
7
16
leaf arrangements
Leaves, petals and seeds arranegment
222.5/137.5 = Φ= 1.618
1/Φ =.618 = Φ -1 , 1- 1/Φ = .382
0·382x360 = 137·5°
1
2
3
4
5
6
7
8
9 10
11
12
13
14
15
16
Benefits:
•Moisture
•Sunlight
•Insects
Leaf arrangements • If we look down on a plant, the leaves
are often arranged so that leaves above do not hide leaves below.
• Each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
• Insects can collect pollens easily.
• In a 120 0 arrangement like a fan, 4th one would hide the first leaf
Why does nature like using Phi in so many plants? • 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.
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.
Pinecones!
Pine cones pa[na kaona
Banana Apple
In vegetable?
5 Red
Spirals
8 Blue
Spirals
1 1.618
WHOLE
WHOLE = LARGE = GOLDEN MEAN
LARGE SMALL
Human Hand
DNA
Beautiful Facial Structures
Dr. Marquadt’s perfect face mask
• Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section, that enigmatic number that has long stood for beauty, and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have.
Leonardo Da Vinci, Mona Lisa
Musical instruments are often based on phi
• Musical scales are based on Fibonacci numbers; piano keyboard scale of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2.
The Bagdad City Gate
Dome of St. Paul: London, England
Windson Castle
Golden Caliper
Advertisement
ART
Architecture
MUSIC
Shape of Universe
Rings of Saturn
Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure.
Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure.
Great Pyramids
Egyptians in the design of the pyramids,
The CN Tower in Toronto, the tallest tower.
The ratio of observation deck at 342 meters to the total height of 553.33 is 0.618, the reciprocal of Phi!
Notre Dame in Paris
Human expectations occur
in a ratio that approaches Phi
• our brains are hardwired to find Fibonacci
progressions naturally pleasing.
• Changes in stock prices largely reflect human opinions, valuations and expectations. 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.
Fibonacci (downward) Retracement with continuing upward trend
• a/b ratios : 23.6%, 38.2%, 50%, 61.8%
• When Fibonacci behavior applies, price “a” may drop 23.6% of “b”, or 38.2%, or 50%, or 61.8%, before it continues its upward trend.
• This is used as an investment guide
b
a 23.6% 38.2% 50%%
61.8%
Φ Is an Infinite Square Root
1 1 1 1 .....
Φ as a Continued Fraction
11
11
11
11 ...
1
Fibonacci numbers define key points in human aging
Huma
n Age
Development Stage
Key Attributes
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
Fibonacci numbers define key points in human aging
Human
Age
Development Stage
Key Attributes
13 Adolescent Thinking, puberty, sexual maturation and drive
21 Young adult
Full physical growth, adult in society, education complete, beginning career, financial responsibility, eligible for voting
34 Mid adult Refinement of adult skills, parenting role
55 Elder adult
Fulfillment of adult skills, serving, retirement begins with eligibility for Medicare, Social Security
89 Completion Insight and wisdom into life
Surely it is logical to conclude that the connections exist because nature, mathematics and the human mind, with its subtle sense of beauty, have one supreme link — they are all the created products of God, the Master Designer
Phi •No rational approximation for long
• e = 2.71828..... ≈ 2+ 5/7 → 7 arms
•π = 3.141592… ≈ 3 + 1/7 → 7 arms
1+ 1
1+ 1
1+ 1
1+ 1
1 + …
8 5
2
2
1
Da Vinci Code
Salary Theorem The less you know, the more you make.
Proof:
• Postulate 1: Knowledge is Power. Postulate 2: Time is Money.
As every engineer knows:
• Power = Work / Time And since
• Knowledge = Power and Time = Money therefore
• Knowledge = Work / Money . Money = Work / Knowledge Thus, as Knowledge approaches zero, Money approaches infinity, regardless of the amount of Work done.
Salary Theorem The less you know, the more you make.
Proof:
• Postulate 1: Knowledge is Power. Postulate 2: Time is Money.
As every engineer knows:
• Power = Work / Time And since
• Knowledge = Power and Time = Money therefore
• Knowledge = Work / Money . Money = Work / Knowledge Thus, as Knowledge approaches zero, Money approaches infinity, regardless of the amount of Work done.
GENERAL EQUATIONS & STATISTICS
• A woman worries about the future until she gets a husband.
• A man never worries about the future until he gets a wife.
• A successful man is one who makes more money than his wife can spend.
• A successful woman is one who can find such a man.
• To be happy with a man, you must understand him a lot and love him a little.
• To be happy with a woman, you must love her a lot and not try to understand her at all.
• Married men live longer than single men, but married men are a lot more willing to die.
• Any married man should forget his mistakes, there's no use in two people remembering the same thing
• SHOPPING MATH
• A man will pay $2 for a $1 item he needs. A woman will pay $1 for a $2 item that she doesn't need.
• Men wake up as good-looking as they went to bed. Women somehow deteriorate during the night.
• A woman marries a man expecting he will change, but he doesn't.
• A man marries a woman expecting that she won't change, and she does.
• A woman has the last word in any argument. Anything a man says after that is the beginning of a new argument.
• There are 2 times when a man doesn't understand a woman - before marriage and after marriage.
ROMANCE MATHEMATICS
• Smart man + smart woman = romance
• Smart man + dumb woman = affair
• Dumb man + smart woman = marriage
• Dumb man + dumb woman = pregnancy
•
OFFICE ARITHMETIC
• Smart boss + smart employee = profit
• Smart boss + dumb employee = production
• Dumb boss + smart employee = promotion
• Dumb boss + dumb employee = overtime
Girl is absolute eveil
AaOrt = baura[-yaa^M
• AaOrt = samaya X Qana
• samaya = Qana Aqaa-t AaOrt = Qana 2
• Qana = baura[-yaaoM ka maUla
(money is the root of all evil)
• AaOrt = Qana 2
• AaOrt = ( baura[-yaaoM ka maUla ) 2
• AaOrt = baura[-yaa^M
SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE THE SUM IS
10000
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• 0050
SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE IS THE SUM 10000?
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SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE IS THE SUM 10000?
• 1000
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• 0030
SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE IS THE SUM 10000?
• 1000
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• 1000
SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE IS THE SUM 10000?
• 1000
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• 0030
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• 1000
SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE IS THE SUM 10000?
• 1000
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• 1000
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SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE IS THE SUM 10000?
• 1000
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SUM THE NUMBERS AS THEY ARE AND NOT DIGIT-VICE IS THE SUM 10000?
Rounding off Psycology
•TOTAL
•10000 OR 9100?