Title: Mathematical Ideas that Shaped the World
1Mathematical Ideas that Shaped the World
2Plan 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?
3Solving 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
4Timeline
- 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.
5Our 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
6Niels 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.
7Career 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.
8A 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
9Evariste 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.
10Galoiss 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
11Symmetry
- What do you think symmetry is?
- Which of these objects do you think is more
symmetric?
12The mathematicians view
- A symmetry is an action that can be performed on
an object to leave it looking the same as before.
13Symmetries 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
14Symmetries of a square
- Can you find all the symmetries of a square?
15Symmetries 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?
16Rules 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.
17Combining 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!
18Multiplication 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
19Homework
- Find the multiplication table for the square!
20The 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
21Symmetry 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
22Symmetries 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!
23Example
- Consider the equation
- (x2 - 5)2 - 24 0
- The roots are
24Symmetries
- 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.
25From 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.
26Motto
- If the symmetries of the solutions can be broken
down into rotations and reflections, then there
is a formula for the solutions.
27Breaking 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
28Breaking 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.
29Breaking 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!
30Abels 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
31No 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.
32A 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.
33Legacy
- 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.
34Back 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!
35Cauchy 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.
36Political turmoil
- In 1830 Paris revolted against Charles X.
- Galois was forced to stay inside his school
despite aching to join the fighting.
37Third 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
38Finally, 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.
39Prison 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
40Love 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.
41More 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?
42The 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.
43Legacy
- 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.
44Wallpaper groups
- The theory of symmetry had far-reaching
consequences for science. - One branch of mathematics looked at the
symmetries of wallpaper.
4517 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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47The 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.
48Sodium chloride (salt)
49Pyrite (iron sulphide)
50Quartz (silicon dioxide)
51Graphite
52Make 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
53Penrose 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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55Where 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
56What 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.