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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. 3

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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 Niels Abel Evariste Galois**

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. 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. Economic crisis between 1807 and 1813 with British blockade of Norway. Much famine and poverty. In 1814 Norway was handed over to Sweden.

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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 ( …) 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. Napoleon abdicates in 1815, Louis XVIII King until 1824, then Charles X.

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Galois’s 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? First and third have rotational but no reflectional symmetry. The middle has 3 rotations and 3 reflections.

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**The mathematician’s 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? A B C 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] All permutations of 3 letters are symmetries of a triangle.

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**Symmetries of a square Can you find all the symmetries of a square? A**

B C D

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**Symmetries of a square A B C D**

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] A B C D Not all permutations of 4 letters are symmetries of a square. E.g. cannot do [A,C,D,B] – keep A fixed and shuffle other letters. Are these all the symmetries of 4 objects?

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Rules of symmetries A group is a mathematician’s 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? A B C 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 Notice the lovely structure of the table: in every row and column each symmetry appears once. Also the red and blue symmetries are in nice blocks.

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**Find the multiplication table for the square!**

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 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…

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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. Let’s do an example!

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Example Consider the equation (x2 - 5) = 0 The roots are

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**Symmetries 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. The equations must have whole numbers or fractions in them. A+B = 2root(2) isn’t allowed.

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**From symmetries to formulae**

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.

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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 See the multiplication table before – the red and blue blocks are distinct. C B

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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. A B C D If we wrote down a massive multiplication table (24 x 24) for the symmetries of 4 objects, it would also break down into nice blocks.

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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! The 60x60 multiplication table here does not break down into nice blocks. It is possible to find 3-fold, 4-fold and 5-fold symmetries, but they are so intertwined that the group does not break down nicely like before. This group is called a ‘simple’ group, somewhat akin to a prime number.

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Abel’s journey From 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 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. Cauchy had also received a 500-page proof from Ruffini, although it had a mistake.

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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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**The term abelian is also named after Abel.**

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. A group is abelian if the order of doing symmetries doesn’t matter.

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Back to Galois 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!

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Cauchy fails again 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. Abel wins the prize post-humously.

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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. The revolution succeeded and the puppet-king Louis-Philippe was put on the throne. This angered some republicans, who didn’t want another king.

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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 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.

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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 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 Rubik’s 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.

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