Transcript Mathematics and aesthetics

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
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Patterns in math – prime
numbers
There 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.
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Let’s try it:
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2=1+1
4=2+2
6=3+3
8=5+3
10 = 5 + 5
12 = 7 + 5
14 = 7 + 7
16 = 13 + 3
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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
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More prime numbers
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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
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From which we get:
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2(2^1) +1 = 5
2(2^2) +1 = 17
2(2^3) + 1 = 257
2(2^4) + 1 = 65537
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These are all primes. So we assume that the next one is, right?
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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
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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
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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
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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?
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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
1 to 100
26
Ratio
1/ln (n)
Deviation (%)
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.
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This law was first discovered empirically.
Meaning, some math geeks sat around and
counted primes and played with logarithms.
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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.
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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
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Euler’s constant
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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]
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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
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0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987
Fibonacci shows up in nature
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Where?
 Rabbit
births
 Honeybees and family trees
 Petals on flowers
 Seed heads
 Pine cones
 Leaf arrangements
Math in art and nature
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The golden ratio (phi) = 1 + [(sqrt(5) –
1)] / 2]
Leonardo Da Vinci
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Uses this proportion
in his artistic work
representing the
body
It shows up in ancient architecture
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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?
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