Mathematical Ideas that Shaped the World

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Mathematical Ideas that Shaped the World

• Symmetry

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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?

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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.

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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.

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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
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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.

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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.

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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

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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.

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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

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Symmetry

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

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The mathematicians view

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

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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
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Symmetries of a square

• Can you find all the symmetries of a square?

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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?

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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.

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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!
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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
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Homework

• Find the multiplication table for the square!

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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

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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

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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!

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Example

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

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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.

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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.

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Motto

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

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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
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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.

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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!
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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

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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.

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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.

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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.

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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!

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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.

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Political turmoil

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

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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

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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.

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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

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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.

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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?

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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.

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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.

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Wallpaper groups

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

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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!

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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.

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Sodium chloride (salt)
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Pyrite (iron sulphide)
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Quartz (silicon dioxide)
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Graphite
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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

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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.

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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

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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.