Lecture 4: Symmetry. Julia Collins. 16th October q 2. q p

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1 Lecture 4: Symmetry Julia Collins 16th October 013 Finding a formula for the quintic The solutions to a quadratic equation ax + bx + c have been taught to school children for centuries: x = b ± b 4ac a The method of finding the solutions was known to the Egyptians and Babylonians 4000 years ago. The solutions to a cubic were discovered much later, around about 1550AD. The discovery of a general formula was the result of much duelling, under-handed acts, plagiarism and surprises, as well as the mathematical brilliance of the men involved (see Lecture 1). For a cubic of the form x 3 + px + q = 0, one solution is x = 3 q + q 4 + p q q 4 + p3 7. (Actually all cubics can be put into this form, so this is a general solution.) The quartic formula was found around the same time as the cubic, and I won t write it down but it involves taking various fourth, third and square roots of combinations of the coefficients of the polynomial. (The coefficients of a polynomial are the numbers in front of the x s, e.g. a, b and c for the quadratic above, and p and q for the cubic.) In general, people wanted to find a formula for the solution of a degree n polynomial. This formula should involve taking n th roots (and other smaller roots) of the coefficients, along with additions, subtractions and simple arithmetic like that. Notice that finding a formula is not the only way to find the solutions. There will always be n (complex) solutions to a degree n polynomial, regardless of whether or not we have a formula to compute them easily. The absence of a formula would not indicate the absence of solutions. 1

2 Symmetries of shapes The symmetries of a shape are those actions which leave it unchanged. How do we write down the symmetries of a triangle? One way is to perform the action and see where the labels of the corners end up. We read off the labels starting with the top one and working clockwise. So the do nothing symmetry is [A, B, C]. Rotation around the centre by 10 leaves the picture looking like this: So this symmetry is [C, A, B]. Similarly, rotation through 40 gives us [B, C, A]. There are also three reflections: [A, C, B] (flipping B and C), [C, B, A] (flipping A and C) and [B, A, C] (flipping A and B). Notice that the collection of symmetries of a triangle, written in this way, give us all possible ways of arranging three objects A, B and C. Similarly, we can write down the symmetries of a cube, reading off the labels from the top left corner and working clockwise. The symmetries are: [A, B, C, D] (do nothing), [D, A, B, C] (rotate by 90 ), [C, D, A, B] (rotate by 180 ), [B, C, D, A] (rotate by 70 ), [B, A, D, C] (flip through a vertical mirror), [D, C, B, A] (flip through a horizontal mirror),

3 [C, B, A, D] (flip A and C through a diagonal mirror) and [A, D, C, B] (flip B and D through the other diagonal mirror). So there are 8 symmetries of a square, but there are 4 ways of ordering 4 objects. For example, there is no symmetry of the square which involves keeping just A fixed and moving around the other 3 corners. To find the remaining symmetries, we must look to another object with 4 corners: the tetrahedron. See if you can find all the missing symmetries! Abel and Galois s solution The massive breakthrough by Abel and Galois was to notice that it was the symmetries of the solution of a polynomial which determined the formula for writing it down. The details can be found in the slides for the lecture (3 9), but the general summary is that if there is an n-fold symmetry to the solutions then there will be an n th root in the formula. A flip is a -fold symmetry; this denotes the presence of square roots. The rotations of a triangle have 3-fold symmetry, so if the 3 solutions of a cubic had this rotational symmetry, then there would be a cube root in the formula. If the solutions of the cubic had triangle symmetry AND flip symmetry, there would be both cube AND square roots in the formula (as seen above). The reason that there is a general formula for solving cubic equations is that the symmetries of 3 objects (i.e. the different ways of ordering 3 objects) break down exactly into flips and rotations. See the multiplication table on slide 19: it breaks down beautifully into corners of blue (flips) and red (rotations). If we were to write down the multiplication table for symmetries of a tetrahedron (i.e. the symmetries of 4 objects) then this would also break down into pretty coloured blocks of flips, triangle rotations and square rotations. The proof that there is no formula for a general quintic comes from looking at the symmetries of 5 objects. These symmetries break down into 60 flips and the 60 different rotational symmetries of a dodecahedron. If we write down the multiplication table for the rotational symmetries of a dodecahedron, we find that there is no way to break it down into nice blocks of smaller symmetries. Sure, you can find 3-fold symmetries of the dodecahedron, but they are so entwined with the other symmetries that you can t isolate them in the table. 3

4 The symmetries of a dodecahedron form what mathematicians call a simple group. It is analogous to a prime number - an object that cannot be broken down into simpler objects. Since the symmetries of 5 objects cannot always be broken down into simpler symmetries, there can be no general formula for a quintic involving 3rd, 4th and 5th roots. However, some quintics have nice symmetries which means that formulae exist - this is what the subject of Galois Theory tells us about. Classification of simple groups It is one of the greatest achievements of 0th century mathematics that we have a list of all the finite simple groups. Yes, we ve found them all. It s not like the prime numbers, where we can keep on finding new and bigger ones that nobody has seen before. It s not that there are only finitely many of them, but that most of the simple groups fall into nice families. The list of finite simple groups involves 18 infinite families of groups together with 6 sporadic groups which don t fit into any of these families. The final group in the list was found in 198 and was called the Monster Group because of its insane size. The Monster Group contains 808, 017, 44, 794, 51, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000 symmetries. The proof of the classification theorem has been written by over 100 different authors in many hundreds of papers, and was only finished in 004. Mathematicians are currently trying to re-write the proof into a more manageable form, but best estimates still think it will take about 5,000 pages. The best place to read about this topic is the book by Marcus du Sautoy called Finding Moonshine: dp/ In the book Marcus describes the journey of all the mathematicians involved in the proof and the surprises they found along the way. The book also covers most of the material in this lecture and contains some nice applications of group theory. Homework problem Write down the multiplication table for the symmetries of a square, like the table for the triangle symmetries on slide 19. There will be 64 elements in the grid to figure out, but the rotation part of it at least will be easy. 4

5 If you are feeling particularly masochistic (or are good at programming) you might like to try finding the multiplication table of the symmetries of a tetrahedron. Coloured in the right way, it should look very pretty! 5