Lecture 4: Symmetry
Julia Collins
16th October 2013
Finding a formula for the quintic
The solutions to a quadratic equation ax2 + bx + c have been taught to school children for
centuries:
√
−b ± b2 − 4ac
x=
2a
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 x3 + px + q = 0, one solution is
s
s
r
r
2
3
3
3
q
q
p
q
q 2 p3
+
+ − −
+ .
x= − +
2
4
27
2
4
27
(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 nth 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.
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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 120◦ leaves the picture looking like
this:
So this symmetry is [C, A, B]. Similarly, rotation through 240◦ 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 270◦ ),
[B, A, D, C] (flip through a vertical mirror), [D, C, B, A] (flip through a horizontal mirror),
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[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 24 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 (23–29), but the general summary is that if there is
an n-fold symmetry to the solutions then there will be an nth root in the formula. A flip is a
2-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.
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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 20th 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 26 ‘sporadic’ groups which don’t fit into any of these families. The final group
in the list was found in 1982 and was called the Monster Group because of its insane size.
The Monster Group contains
808, 017, 424, 794, 512, 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 2004. 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:
http://www.amazon.co.uk/Finding-Moonshine-Mathematicians-Journey-Symmetry/
dp/0007214626
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.
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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!
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