A tetrahedron can be placed in 12 distinct positions by rotation
alone. These are illustrated above
in the cycle graph format, along with the 180° edge (blue arrows) and 120°
vertex (reddish arrows) rotations that permute the tetrahedron through the
positions. The 12 rotations form the rotation (symmetry) group of the figure.
By Marcus du Sautoy
University of Oxford
Edited by Andy Ross
Symmetry is a central concept in both science and the arts.
Symmetry has unlocked the secrets of the fundamental particles that make up the
The saga begins with the extraordinary revelation of Evariste Galois in 1832
that just as numbers can be built out of the indivisible primes, symmetrical
objects too can be decomposed into indivisible symmetrical objects. Christened
simple groups, these symmetrical objects are the atoms of the world of symmetry.
For mathematicians the symmetries of a shape are all the ways I can rearrange
the shape and place it back down inside an outline of the shape so that the
shape looks like it did before I moved it. Each symmetry of a shape is bit like
a magic trick move: look away and I move the shape but when you look back it's
as if the shape hasn't moved. For example, I can rotate a 15-sided polygon by a
15th of a turn and it looks just the same.
Galois understood that the symmetries of this 15-sided polygon can be built out
of the symmetries of two smaller shapes sitting inside the large shape, namely a
pentagon and a triangle. How can you rotate the 15-sided polygon through a
fifteenth of a turn using the rotations of the pentagon and triangle? First
rotate the pentagon by two fifths of a turn; then pull back in the opposite
direction by rotating the triangle by a third of a turn. The combined effect is
a rotation of a 15th of turn.
So the group of symmetries of the 15-sided polygon are built out of the
symmetries of a pentagon and a triangle. But the rotations of these prime sided
shapes cannot be broken down. So just as prime numbers are the building blocks
of all numbers it turns out that prime sided shapes are some of the first
building blocks of the world of symmetry.
One of the achievements for which John Thompson won the Abel Prize in 2008 is a
theorem he proved with the late Walter Feit. Together they proved that if an
object or structure has an odd number of symmetries then its symmetries can be
broken down into the symmetries of these prime sided shapes. Called the Odd
Order Theorem, its proof was published in 1963 and ran to 255 pages.
There are other shapes that cannot be broken down so easily. For example, take
the classic football made up of pentagons and hexagons. This shape has 60
rotational symmetries that rearrange the shape so that all the pentagons and
hexagons line up again. Galois proved that the 60 rotations of the football are
as indivisible as if it were a prime sided shape. The rotations of a pentagon
are a subset of the symmetries of the football, but there are no shapes whose
symmetries can be combined with those of the pentagon to give all the symmetries
of the football.
There are many other shapes whose symmetries are indivisible. For example, the
symmetries of hypercubes in higher dimensions are behind one of 16 new families
of Lie groups. For unlocking the secrets of these groups, Jacques Tits won the
Abel Prize together with Thompson. Tits constructed geometrical settings in
higher dimensions which help explain the symmetries of these new families.
But then French mathematician Emile Mathieu discovered five symmetrical shapes
that didn't seem to fit into any of these families of groups. And in 1965,
Croatian mathematician Zvonimir Janko claimed to have discovered a new
indivisible symmetry, a sixth sporadic group.
Janko's discovery was the beginning of a crazy period in the story of symmetry
where mathematicians discovered a whole range of strange indivisible sporadic
groups. Many of the discoveries depended on using a formula developed by
Thompson to predict how many symmetries such a sporadic group might have.
Both Thompson and Tits have their names attached to some of the sporadic
groups of symmetries that appeared over the decades since Janko's discovery. The
culmination of this period of exploration was the discovery of a 26th object
that sits in a space of 196,883 dimensions and has more symmetries than there
are atoms in the sun. It is called the Monster and is the largest of the
sporadic groups. But we think there are no more indivisible symmetries.
Thanks to the work of mathematicians like Thompson and Tits, we believe we now
have a complete list of the building blocks of symmetry. This is one of the
greatest achievement in the history of mathematics.
Grand designs: Symmetry's hidden depths
By Marcus du Sautoy
New Scientist, June 11, 2008
University of Oslo, May 20: Norwegian King Harald (right) presents the
Abel Prize for 2008
to John Thompson (left) and Jacques Tits (center) for their work in group theory.
John Griggs Thompson, 75, teaches at the University of Florida. He was born in
Ottawa, Kansas, and graduated from Yale University in 1955, received his
doctorate from the University of Chicago in 1959. He taught at Harvard
University and then at the University of Chicago, before moving to Britain,
where he spent more than 20 years teaching at the University of Cambridge.
Jacques Tits, 77, is professor emeritus at College de France in Paris. Born in
Brussels, he was admitted to the Free University of Brussels at age 14, and
received his doctorate at the age of 20. He taught there (and at the University
of Bonn in 1964) before he accepted the chair of group theory at the College de
France 1973 and became a French citizen in 1974. He retired in 2000.
Finite Group Representations
AR This is a topic that
amply deserves more extended study, if only to appreciate more fully the work of