Mathematical Ideas that
Shaped the World
Symmetry
Plan for this class
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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?
Solving equations
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A polynomial equation has the form
2x3 – 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.
Timeline
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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.
Our heros
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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
Niels Abel (1802 – …)
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Born in south-east
Norway during troubled
economic and political
times.
Wasn’t 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.
Career beginnings
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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.
A promise
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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.
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Sends the paper to Cauchy…
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Evariste Galois (1811 - …)
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Born in Bourg-la-Reine, on
the outskirts of Paris, also
during turbulent times.
Age 12, goes to the Lycée
Louis-la-Grand, a prisonlike school.
Age 14 reads a maths book
in 2 days, though it would
normally take 2 years to
teach.
Galois’s dream
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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…
Symmetry
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What do you think symmetry is?
Which of these objects do you think is more
symmetric?
The mathematician’s view
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A symmetry is an action that can be
performed on an object to leave it looking
the same as before.
Symmetries of an equilateral triangle
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How do we write down all the
symmetries of a triangle?
A
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]
C
B
Symmetries of a square
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Can you find all the symmetries of a square?
A
B
D
C
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]
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F4: [C,B,A,D]
A
B
D
C
Are these all the symmetries of 4 objects?
Rules of symmetries
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A group is a mathematician’s word for a
collection of symmetries.
A collection of symmetries must follow these
rules:
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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.
Combining triangle symmetries
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What happens if we take our triangle and do
a rotation followed by a reflection? Is this
one of the other symmetries?
A
Rotation R1 : [C,A,B]
followed by
Reflection F1 [A,C,B]
gives [C,B,A] – F2!
C
B
Multiplication table of symmetry
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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
N
R1 R2 F1 F2 F3
N
R1 R2 F1 F2 F3
R1 R1 R2 N
R2 R2 N
F2 F3 F1
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
Homework
Find the multiplication table
for the square!
The integers as a group
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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…
Symmetry is the key!
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Abel and Galois’s 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…
Symmetries tell you the formula
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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.
Let’s do an example!
Example
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Consider the equation
(x2 - 5)2 - 24 = 0
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The roots are
Symmetries
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Some equations are
 A + D = 0
 B + C = 0
 (A+B)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.
From symmetries to formulae
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If you do a reflection twice, it’s 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.
Motto
If the symmetries of the solutions
can be broken down into rotations
and reflections, then there is a
formula for the solutions.
Breaking down groups - cubics
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The symmetries of 3 objects are all the
symmetries of a triangle.
These can always be broken down into
reflections and rotations.
A
C
B
Breaking down groups - quartics
The symmetries of 4 objects are the
symmetries of a tetrahedron. (Not always a
square!)
A
 These can be broken down
into rotations of triangles,
D
B
rotations of squares, and
reflections.
C
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Breaking down groups - quintics
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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!
Abel’s journey
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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…
No joy
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Unfortunately, Cauchy had
neglected to present Abel’s
paper to the Academy, and
hadn’t 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.
A tragic end
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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.
Legacy
In 2003 the Abel prize was set up by
the Norwegian Academy. It’s like a
Nobel prize for mathematicians, worth
£500,000.
The term abelian is also named after
Abel.
Back to Galois
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In 1828, the young Galois is also waiting for a
reply from Cauchy.
Meanwhile, in 1829, Galois’s 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!
Cauchy fails again
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Forced to attend a lesser
institution, Galois’s 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.
Political turmoil
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In 1830 Paris
revolted against
Charles X.
Galois was
forced to stay
inside his
school despite
aching to join
the fighting.
Third time lucky?
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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…
Finally, a reply!
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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.
Prison life
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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…
Love lost and found
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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.
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
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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.
Legacy
In 1843 Galois’s 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.
Wallpaper groups
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The theory of symmetry had far-reaching
consequences for science.
One branch of mathematics looked at the
symmetries of wallpaper.
17 types of wallpaper
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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!
The next dimension
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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.
Sodium chloride (salt)
Pyrite (iron sulphide)
Quartz (silicon dioxide)
Graphite
Make a crystal – win a Nobel prize!
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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…
Penrose tilings
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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.
Where else are groups?
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Group theory is now found in all aspects of
our modern lives, including
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Cryptography in credit cards and banking
Getting a brain scan
Listening to digital music (and video)
Bar codes
Puzzles like the Rubik’s cube
Analysing viruses like HIV and herpes
What did we learn?
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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.