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.

http://www.abc.net.au/science/photos/mathsinnature/photo5.htm

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

http://ccins.camosun.bc.ca/~jbritton/fibslide/fib12.gif

<>

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.

http://ccins.camosun.bc.ca/~jbritton/fibslide/fib30.gif

http://ccins.camosun.bc.ca/~jbritton/fibslide/fib31.gif

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

सर्व धमव समभार् Symbol

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.

Golden mean gauge

The Bagdad City Gate

Dome of St. Paul: London, England

Windson Castle

Golden Caliper

Advertisement

ART

Architecture

MUSIC

Shape of Universe

http://home.aanet.com.au/robertw/Stella.html

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

http://www.mathematicianspictures.com/PHI_GOLDEN_RATIO_GOLDEN_MEAN/Phi_Golden_Ratio_Golden_Mean.htm

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

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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?

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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?

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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?

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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?