‘The Code’ – by Prof. Marcus du Sautoy

Ever since humans evolved on this planet we have been trying

to make sense of the world around us.

We have attempted to explain why the world looks and

behaves the way it does, to predict what the future holds. And

in our search for answers we have uncovered a code that

makes sense of the huge complexity that confronts us -

mathematics.

By translating nature into the code of numbers we have

revealed hidden structures and patterns that control our

environment.

But not only that. By tapping into nature's code we have been

able to change our surroundings, have built extraordinary

cities, and developed amazing technology that has resulted in

the modern world.

Buzzing quietly beneath the planet we inhabit is an unseen

world of numbers, patterns and geometry. Mathematics is the

code that makes sense of our universe.

Some Basic Mathematical Relationships.

(Not to be memorized)

a x a = a2 a x a x a = a3 > Greater than Less than <

Properties of a Circle.

Circumference 𝐶 = 2𝜋r Area 𝐴 = 𝜋𝑟2

Trigonometrical Properties.

𝑎2 + 𝑏2 = 𝑐2

Area of triangle = a x b/2

Sinθ = Opposite/Hypotenuse = b/c

Cosθ = Adjacent/Hypotenuse = a/c

Tanθ = Opposite/Adjacent = b/a

a

b

c θ

Basic Relationships (Continued)

h

a b

A B

Area A = axh/2 & Area B = bxh/2

Area A+B = ah/2 + bh/2 = hx(a+b)/2

Area of any triangle =½ x base x height

h

Volume of Cone =

1/3xBase Area xh

= πr2 x h/3 Vol of Sphere = 4/3x πr3

r

Pythagoras

3

4

c

For a right angled triangle. a2+b2=c2

Area =16

Area=9

Area=25

16+9=25

The same result applies to all right angled triangles.

C=5

2xR

A=2R x 2R

Area=4R2

R

X

X2+X2=R2

2X2=R2

Triangle area

=X2/2

a= 8xX2/2

a=4xX2

area = 2R2

Large Square

Small Square

Compare the areas of the Large & Small Squares.

ie The large square is exactly twice the area

of the small square.

x

C=2πr - Find a Value for Pi (π)

r

r

x

y

30o

30o

Perimeter of large hexagon, P=12x Perimeter of small hexagon, p=12y

X=rxtan300 y=rsin300

X=0.5773xr y=0.5xr

P=12x = 6.927r

π < 3.4638 p=12y = 6.0r

π > 3.00

3.00 < π < 3.4638

Pi (π) {continued}

The same principle can be applied using polygons of ever increasing sides.

Then the perimeter length of the polygon will get closer & closer to the circle

circumference

For Polygons with n sides;

p=2xnxrxSin(180/n) & P=2xnxrxTan(180/n)

Or π > nxSin(180/n) π < nxTan(180/n)

For a 96 sided polygon 3.140 < π < 3.146

For a 360 sided polygon 3.14155 < π < 3.14167

The exact value of Pi to 5 decimal places is 3.14159, so we are

getting closer with each step.

Area of a Circle Radius r

Divide the entire circle into a

series of thin triangular slices.

Assemble the slices into a

rectangular form as below. r

C/2 = Π x r

r

Area of Rectangle & Circle = r x πr = πr2

• A prime number can be divided, without a remainder, only by itself and by 1.

For example, 17 can be divided only by 17 and by 1.

•

Some facts:

• The only even prime number is 2. All other even numbers can be divided by 2.

• If the sum of a number's digits is a multiple of 3, that number can be divided by

3.

• No prime number greater than 5 ends in a 5. Any number greater than 5 that

ends in a 5 can be divided by 5.

• Zero and 1 are not considered prime numbers.

• Except for 0 and 1, a number is either a prime number or a composite number.

A composite number is defined as any number, greater than 1, that is not

prime.

• To prove whether a number is a prime number, first try dividing it by 2, and see

if you get a whole number. If you do, it can't be a prime number. If you don't

get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is

divisible by 3) and so on, always dividing by a prime number (see table below).

Prime Numbers

Prime Numbers (Continued)

• Here is a table of all prime numbers up to 1,000:

2 3 5 7 11 13 17 19 23

29 31 37 41 43 47 53 59 61 67

71 73 79 83 89 97 101 103 107 109

113 127 131 137 139 149 151 157 163 167

173 179 181 191 193 197 199 211 223 227

229 233 239 241 251 257 263 269 271 277

281 283 293 307 311 313 317 331 337 347

349 353 359 367 373 379 383 389 397 401

409 419 421 431 433 439 443 449 457 461

463 467 479 487 491 499 503 509 521 523

541 547 557 563 569 571 577 587 593 599

601 607 613 617 619 631 641 643 647 653

659 661 673 677 683 691 701 709 719 727

733 739 743 751 757 761 769 773 787 797

809 811 821 823 827 829 839 853 857 859

863 877 881 883 887 907 911 919 929 937

Prime numbers in Nature

Every 13 years, in Tenessee, for six weeks there is a chorus of an insect that fascinates mathematicians. Only found in the eastern areas of North America, this cicadas survival depends on exploiting the strange properties of some of the most fundamental numbers in mathematics - the primes, numbers divisible only by themselves and one.

The cicadas appear periodically but only emerge after a prime number of years. In the case of the brood appearing around Nashville this year, 13 years. The forests have been quiet for 12 years since the last invasion of these mathematical bugs in 1998 and they won't appear again until 2024.

This choice of a 13-year cycle doesn't seem too arbitrary. There are another two broods across north America that also have this 13-year life cycle, appearing in different regions and different years. In addition there are another 12 broods that appear every 17 years.

Prime numbers in Nature (Continued)

You could just dismiss these numbers as random. But it's very curious that there are no cicadas with 12, 14, 15, 16 or 18-year life cycles. However look at these cicadas through the mathematician's eyes and a pattern begins to emerge.

Because 13 and 17 are both indivisible this gives the cicadas an evolutionary advantage as primes are helpful in avoiding other animals with periodic behaviour. Suppose for example that a predator appears every six years in the forest. Then a cicada with an eight or nine-year life cycle will coincide with the predator much more often than a cicada with a seven-year prime life cycle.

These insects are tapping into the code of mathematics for their survival. The cicadas unwittingly discovered the primes using evolutionary tactics but humans have understood that these numbers not just the key to survival but are the very building blocks of the code of mathematics.

Prime numbers in Nature (Continued)

A large number of plants also utilise prime numbers in their development. Count the number of petals on common flowers, many of them have prime number arrangements.

Similarily, the budding or growing points on certain shrubs follow a prime number configuration around the circumference of the branch.

Prime numbers in Cryptography The cryptography that keeps our credit cards secure when we shop online exploits the same numbers that protect the cicadas in North America - the primes.

Every time you send your credit card number to a website your are depending on primes to keep your details secret. To encode your credit card number your computer receives a public number N from the website, which it uses to perform a calculation with your credit card number.

This scrambles your details so that the encoded message can be sent across the internet. But to decode the message the website uses the primes which divide N to undo the calculation. Although N is public, the primes which divide N are the secret keys which unlock the secret.

The reason this is so secure is that although it is easy to multiply two prime numbers together it is almost impossible to pull them apart.

An Exercise in Logic

3 men are sharing a pizza. A takes a half and B & C in turn take half of what is left. They continue taking half of the residual pie until none is left.

If they share the cost of £4.20 in proportion to their share of the pizza, what does each pay?

If A has a share p, B has p/2 & C has a p/4.

p+p/2+p/4 = 1 (4p+2p+p)/4 = 1 7p=4 p=4/7

A B

C

A

A pays £4.20 x 4/7 = £2.40

B pays £4.20 x 2/7 = £1.20

C pays £4.20 x 1/7 = £0.60

THAT’S IT FOLKS!