Mathematics, patterns, nature, and aesthetics

Math is beautiful, elegant

Consider the tidiness of proofs about concepts

How beautifully science uses math to explain the world

Patterns in math – prime numbersThere is something about prime numbers and the

nature of math that is endlessly interesting. Let’s look as some discoveries to see why.

Goldbach’s conjecture Goldbach was a mathematician who claimed

that every even number could be demonstrated to be a sum of two prime numbers.

Let’s try it:

2 = 1 + 1 4 = 2 + 2 6 = 3 + 3 8 = 5 + 3 10 = 5 + 5 12 = 7 + 5 14 = 7 + 7 16 = 13 + 3

Goldbach

We could go on doing this for a long time. Indeed, using computers mathematicians have proven this for every even number up to 100,000,000,000,000.

But they have found no way to prove Goldbach’s conjecture true.

No deductive rigorous proof yet accepted by the mathematical community

More prime numbers

Many mathematicians have tried to figure out formulas that produce only prime numbers. Fermat – who we will come back to – devised this formula:

22^n + 1 = prime number

  From which we get:

2(2^1) +1 = 52(2^2) +1 = 17

2(2^3) + 1 = 257 2(2^4) + 1 = 65537

 

These are all primes. So we assume that the next one is, right?

Next up: 2(2^5) + 1 = 4,294,697,297

A prime number? Seems so. But Euler – you’ve probably heard of him – using just his intuition (no calculators or computers at his time) figured out that the latter number can be arrived at by multiplying:

 

6,700,417 and 641

  This shows Euler’s capability given there were no computing machines at

this time. This kind of lesson teaches us not to jump to conclusions using induction.

Others

There are others equally as tricky. Consider:n2 – n + 41

 This gives primes up to 40, but fails on 41. Interesting.

Another one:

  n2 – 79n + 1601

You guessed it – it works up to 79 but fails at 80.

These kind of tricks are more easily dispelled now-a-days. But they weren’t in the past.

Logarithms and prime numbers

Another question that number theorists wrestled with is this: is there any way to represent mathematically the diminishing percentage of prime numbers among very large numbers?

There is, indeed. Here is the law:

The percentage of prime numbers within an interval from 1 to any large number (n) is approximately stated by the natural logarithm of n.

Demonstrated:Interval 1 to n

Number of primes

Ratio 1/ln (n) Deviation (%)

1 to 100 26 0.260 0.217 20

1 to 1000 168 0.168 0.145 16

1 to 106 78498 0.078498

0.072382 8

1 to 109 50847478 0.0508 0.048254942

5

You’ll see that column three (n divided by the number of primes from 1 to n) becomes closer and closer to the reciprocal of the natural logarithm of n.

This law was first discovered empirically. Meaning, some math geeks sat around and counted primes and played with logarithms.

Unlike in Goldbach’s case, however, soon before the turn of the twentieth century French mathematicians Hadamard and Belgian de la Vallée Poussin proved it.

I won’t it explain it here because I have no idea how to, but it is nevertheless a remarkably interesting discovery.

Buffon’s needle problem

Divide a paper with parallel lines one unit apart

Drop a pin unit long The probability it crosses one of the

parallel lines is 2/pi

Euler’s constant

Euler’s constant, or e, can be arrived at using infinite series of factorials:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! ...

e does some interesting things in math. If you’ve studied calculus, you know that the integral of ex is ex:

∫ex = ex +c

(For non-math folks, the C is just a constant that could mean anything. For all intents and purpose, the integral of ex is itself.)

Likewise, the derivative of ex is also ex:

(ex)’ = ex

Numbers do interesting things. But this is just pure math, right?

Well, no – e shows up all of the time in study of the natural world. You need it to explain things such as radioactive decay (which we use to know how old things on the Earth actually are), the spread of epidemics, compound interest, and population

More Euler

We could say that these are the five most important numbers in math:

 

e, π, 1, 0, and i [or √(-1), the imaginary number]

Euler discovered this equation:

eiπ + 1 = 0

“What can be more mystical than an imaginary number interacting with real numbers [that show up everywhere in the world] to produce nothing?”

Fibonacci sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

Fibonacci shows up in nature

Where?Rabbit birthsHoneybees and family treesPetals on flowersSeed headsPine conesLeaf arrangements

Math in art and nature

The golden ratio (phi) = 1 + [(sqrt(5) – 1)] / 2]

Leonardo Da Vinci

Uses this proportion in his artistic work representing the body

It shows up in ancient architecture

Parthenon

Find the ratios between consecutive numbers of the fibonacci sequence

Is there not something beautiful, even spiritual, about all this?

How do we explain it?