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Mathematical Ideas that Shaped the World

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Title: Mathematical Ideas that Shaped the World


1
Mathematical Ideas that Shaped the World
  • Symmetry

2
Plan for this class
  • To learn about the tragic lives of Niels Abel and
    Evariste Galois
  • What is symmetry? How do mathematicians think
    about symmetry?
  • What has a dodecahedron got to do with finding
    solutions of equations?
  • Why has the mathematics of symmetry been so
    important in modern day life?

3
Solving equations
  • A polynomial equation has the form
  • 2x 5x2 3x 1 0
  • The highest power of x is called the degree of
    the equation.
  • A degree 2 polynomial is called quadratic.
  • A degree 3 polynomial is called cubic.
  • A degree 4 polynomial is called quartic.
  • A degree 5 polynomial is called quintic.

3
4
Timeline
  • The solution of quadratics was found 4000 years
    ago by the Babylonians.
  • The solution of cubics and quartics were found
    450 years ago by Cardano, Tartaglia and Ferrari.
  • By 1800, there was still no general formula to
    solve the quintic. It was one of the greatest
    unsolved problems of the time.

5
Our heros
  • At the beginning of the 19th century, two men
    were born whose quest for the solution of the
    quintic changed mathematics forever

Niels Abel
and
Evariste Galois
6
Niels Abel (1802 )
  • Born in south-east Norway during troubled
    economic and political times.
  • Wasnt inspired by maths until his school got a
    new teacher, Holmboë, who told him about the big
    unsolved maths problems.
  • In 1820 his father died in disgrace, leaving no
    money and 5 children for Abel to look after.

7
Career beginnings
  • Holmboë pays for Abel to finish his schooling and
    go to university.
  • Abel starts work on finding a quintic formula,
    and thinks he has a solution!
  • Corresponds with Degen, the leading Nordic
    mathematician. Finds a mistake in his solution,
    but is invited to Copenhagen to work further with
    Degen.

8
A promise
  • In 1824, Abel falls in love with Christine Kemp,
    but has no money to afford marriage. Promises to
    find a professorship and come back for her
  • The same year, he finds a solution for the
    quintic. Has to write it down in 6 pages, since
    that was all the paper he could afford.
  • Sends the paper to Cauchy

9
Evariste Galois (1811 - )
  • Born in Bourg-la-Reine, on the outskirts of
    Paris, also during turbulent times.
  • Age 12, goes to the Lycée Louis-la-Grand, a
    prison-like school.
  • Age 14 reads a maths book in 2 days, though it
    would normally take 2 years to teach.

10
Galoiss dream
  • Age 16, he takes the entrance exam for the École
    Polytechnique, the most prestigious institution
    for mathematics in France.
  • He fails.
  • Begins to work on the quintic formula and sends
    his first ideas to Cauchy in the hope of winning
    a place at the École

11
Symmetry
  • What do you think symmetry is?
  • Which of these objects do you think is more
    symmetric?

12
The mathematicians view
  • A symmetry is an action that can be performed on
    an object to leave it looking the same as before.

13
Symmetries of an equilateral triangle
  • How do we write down all the symmetries of a
    triangle?

Rotations Reflections
N A,B,C F1 A,C,B
R1 C,A,B F2 C,B,A
R2 B,C,A F3 B,A,C
14
Symmetries of a square
  • Can you find all the symmetries of a square?

15
Symmetries of a square
Rotations Reflections
N A,B,C,D F1 B,A,D,C
R1 D,A,B,C F2 D,C,B,A
R2 C,D,A,B F3 A,D,C,B
R3 B,C,D,A F4 C,B,A,D
  • Are these all the symmetries of 4 objects?

16
Rules of symmetries
  • A group is a mathematicians word for a
    collection of symmetries.
  • A collection of symmetries must follow these
    rules
  • There is a symmetry which is do nothing.
  • Every symmetry can be undone by another symmetry.
  • Doing one symmetry followed by another is the
    same as doing one of the other symmetries.

17
Combining triangle symmetries
  • What happens if we take our triangle and do a
    rotation followed by a reflection? Is this one of
    the other symmetries?

Rotation R1 C,A,B followed
by Reflection F1 A,C,B gives
C,B,A F2!
18
Multiplication table of symmetry
  • To get a full picture of the triangle symmetries,
    we write down a multiplication table of how the
    symmetries interact.
  • Notice that it matters what order we do the
    multiplication!

N R1 R2 F1 F2 F3
N N R1 R2 F1 F2 F3
R1 R1 R2 N F2 F3 F1
R2 R2 N R1 F3 F1 F2
F1 F1 F3 F2 N R2 R1
F2 F2 F1 F3 R1 N R2
F3 F3 F2 F1 R2 R1 N
19
Homework
  • Find the multiplication table for the square!

20
The integers as a group
  • What are the symmetries of the number line?
  • Answer translations left and right!
  • Each number is itself an action. E.g. 2 means
    shift right by 2 units.
  • The do nothing symmetry is
  • The undo symmetry is
  • Doing one symmetry followed by another is

21
Symmetry is the key!
  • Abel and Galoiss massive breakthrough (which
    they had independently) was that the symmetries
    of the solutions of an equation were the key to
    writing down a formula.
  • And that it all came down to the symmetries of a
    dodecahedron

22
Symmetries tell you the formula
  • A polynomial of degree n always has n solutions
    (although some of them may be imaginary/complex).
  • We can write down all the equations that describe
    the solutions.
  • We then see which symmetries of the solutions
    still make these equations true.
  • Lets do an example!

23
Example
  • Consider the equation
  • (x2 - 5)2 - 24 0
  • The roots are

24
Symmetries
  • Some equations are
  • A D 0
  • B C 0
  • (AB)2 8
  • The symmetries preserving these equations are
    A,B,C,D, B,A,D,C, C,D,A,B, D,C,B,A.
  • These are reflection symmetries.

25
From symmetries to formulae
  • If you do a reflection twice, its the same as
    doing nothing. The fact that we get reflections
    in the last example means that square roots will
    appear in the solution of the polynomial.
  • If there had been some rotational symmetry, we
    would have found we need some 4th roots to get
    the solution.

26
Motto
  • If the symmetries of the solutions can be broken
    down into rotations and reflections, then there
    is a formula for the solutions.

27
Breaking down groups - cubics
  • The symmetries of 3 objects are all the
    symmetries of a triangle.
  • These can always be broken down into reflections
    and rotations.

A
B
C
28
Breaking down groups - quartics
  • The symmetries of 4 objects are the symmetries of
    a tetrahedron. (Not always a square!)
  • These can be broken down
  • into rotations of triangles,
  • rotations of squares, and
  • reflections.

29
Breaking down groups - quintics
  • The symmetries of 5 objects break down into
    reflections and the 60 symmetries of a
    dodecahedron.
  • Abel proved that the dodecahedron symmetries do
    not break down into anything smaller.

This means that some quintics have no formula!
30
Abels journey
  • From 1825 - 1827 Abel was travelling around
    Europe telling everyone about his wonderful
    discovery.
  • But he was a shy, modest man and needed some
    influential friends.
  • When he got to Paris, he hoped for a warm welcome
    from Cauchy, who had received his paper

31
No joy
  • Unfortunately, Cauchy had neglected to present
    Abels paper to the Academy, and hadnt even read
    it himself.
  • A second paper is also ignored by Cauchy.
  • Dejected and poor, Abel returns to Norway with no
    money and no job.

32
A tragic end
  • In December 1828 Abel spends Christmas with his
    fiancée, but has no money for warm clothes. After
    a romantic sleigh ride he falls ill.
  • His friends notice his plight and plead with the
    King of Sweden to get him a position.
  • On 8th April he is offered a job at the
    University of Berlin
  • but the letter arrives a day too late.

33
Legacy
  • In 2003 the Abel prize was set up by the
    Norwegian Academy. Its like a Nobel prize for
    mathematicians, worth 500,000.
  • The term abelian is also named after Abel.

34
Back to Galois
  • In 1828, the young Galois is also waiting for a
    reply from Cauchy.
  • Meanwhile, in 1829, Galoiss father commits
    suicide and political violence breaks out at the
    funeral.
  • 2 days later, Galois retakes the exam to get into
    the École Polytechnique but fails again, even
    throwing a board rubber at the examiners!

35
Cauchy fails again
  • Forced to attend a lesser institution, Galoiss
    hopes are all on Cauchy now
  • Cauchy loses the manuscript.
  • Galois re-submits a new one, hoping to win the
    Grand Prix prize in mathematics.
  • His new referee, Fourier, dies before reading the
    manuscript and Galois is never considered for the
    prize.

36
Political turmoil
  • In 1830 Paris revolted against Charles X.
  • Galois was forced to stay inside his school
    despite aching to join the fighting.

37
Third time lucky?
  • After accusing the headmaster of treason, Galois
    is expelled.
  • He joins 2 militant Republican group, both of
    which become outlawed.
  • To earn money, he gives public lectures about his
    work. The famous mathematician Poisson invites
    Galois to submit his manuscript a third time.
  • Several months pass and still nothing

38
Finally, a reply!
  • At a banquet of one of his secret societies,
    Galois holds a dagger and raises a toast to the
    new king Louis Philippe.
  • He is arrested and tried, but acquitted of
    plotting to kill the King.
  • His friends blame Poisson for his actions.
  • Poisson retaliates by condemning the paper as
    unclear and incomplete.

39
Prison life
  • In his despair, Galois turns to politics again. A
    week later he is arrested and sentenced to 9
    months in prison.
  • Re-writes much of his manuscript and makes the
    first definition of a group.
  • In Spring 1832 he is moved to a new prison
    because of a cholera epidemic. A month later he
    is free, and meets the daughter of the local
    doctor

40
Love lost and found
  • Stéphanie is initially taken with Galois, but he
    is inept at building a relationship.
  • Eventually his advances are rebuffed.
  • He is again distraught and starts attending
    secret political meetings.

41
More tragedy
  • On 30 May 1832, Galois is found lying in a field
    with a single gunshot wound to his stomach. He
    dies the next day.
  • In a letter to a friend, he wrote
  • I beg patriots, my friends, not to reproach me
    for dying otherwise than for my country. I die
    the victim of an infamous coquette and her two
    dupes. It is in a miserable piece of slander that
    I end my life. Oh! Why die for something so
    little, so contemptible?

42
The night before
  • The night before the duel, Galois wrote up the
    last of his mathematical ideas and asked his
    friend Chevalier to send his papers to the best
    mathematicians across Europe.

43
Legacy
  • In 1843 Galoiss papers were finally read and
    published. His work laid the foundations of
    modern Group Theory, and spawned a whole new
    branch of mathematics which is now called Galois
    Theory.
  • All before the age of 21.

44
Wallpaper groups
  • The theory of symmetry had far-reaching
    consequences for science.
  • One branch of mathematics looked at the
    symmetries of wallpaper.

45
17 types of wallpaper
  • You might think there are endless designs of
    wallpaper to choose from, but actually there are
    only really 17!
  • The result was proved in 1891 by Evgraf Fedorov,
    a Russian mathematician and crystallographer.
  • All 17 designs were discovered by the ancient
    Egyptians and Muslims go visit the Alhambra
    palace in Granada, Spain!

46
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47
The next dimension
  • How many symmetric structures are there in 3
    dimensions?
  • Answer There are 230, and they are known to
    chemists as crystals!
  • We can often see the atomic crystal symmetry by
    looking at the macroscopic shape of the crystal.

48
Sodium chloride (salt)
49
Pyrite (iron sulphide)
50
Quartz (silicon dioxide)
51
Graphite
52
Make a crystal win a Nobel prize!
  • Scientists use crystal structures to engineer new
    materials with special properties.
  • For example, the creation of graphene won the
    Nobel prize in 2010. It is a hexagonal lattice of
    atoms which is the strongest substance ever
    found.
  • It was quasicrystals which won the Chemistry
    Nobel Prize in 2011 too

53
Penrose tilings
  • A quasicrystal is unlike normal crystals. It is
    made of 2 different shapes rather than one.
  • The pattern of these two shapes may never repeat.
    The phenomenon was first discovered by
    mathematicians in the 1970s.
  • They were called Penrose tiles.

54
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55
Where else are groups?
  • Group theory is now found in all aspects of our
    modern lives, including
  • Cryptography in credit cards and banking
  • Getting a brain scan
  • Listening to digital music (and video)
  • Bar codes
  • Puzzles like the Rubiks cube
  • Analysing viruses like HIV and herpes

56
What did we learn?
  • That the solutions to seemingly useless
    mathematical problems can have far-reaching
    consequences.
  • That Cauchy was not a very nice person and
    inadvertently caused the premature deaths of two
    brilliant young mathematicians.
  • That the mathematical notion of symmetry is
    integral to modern physics, chemistry, wallpaper
    design and technology.
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