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Part 1: Symmetry
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| 1: What is symmetry? | Intuitively,
you could say that the letter Z has "the same amount of symmetry" as the
letter N , but "more symmetry" than the letter R and "less symmetry" than
the letter H. The mathematical view of symmetry is that it is a "dynamic"
rather than a "static" concept. That is, the more
transformations there are of an object into itself, the more symmetric it is.
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| 2: Group theory. | Groups are used to measure the symmetry of objects. For example, in 2GA2 you learnt about the cyclic group Cn which describes the rotations or rigid motions of the regular n-gon and the dihedral group which describes all the symmetries of the regular n-gon. | . |
| 3:Regular polyhedra. | Note
that since there are infinitely many cyclic and dihedral groups, there
must be infinitely many regular polygons. It may surprise you to learn
that there are only five regular polyhedra, the three dimensional analogues
of the regular polygons. These are the Platonic solids discovered
by the Pythagoreans whose symmetries are described in Book 12 of Euclid's
Elements. They are
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| 4. History of groups. | The
Ancient Greeks did not use group theory to prove this, but nowadays we
use groups to construct the Platonic solids and show there are no other
regular polyhedra. Incidentally, we use the same methods to classify the
n-dimensional analogues of the regular polygons and regular polyhedra,
considered as symmetric objects in Rn.
The use of groups to measure the symmetries of geometric objects is really an afterthought. Groups were originally invented by the 21-year old French mathematician Evariste Galois in 1832 to describe the symmetries of the roots of a polynomial with real coefficients. This is rather difficult to describe in this introduction to the course, but basically it means that if you consider the roots as complex numbers they actually lie in a subfield of the complex field and automorphisms of this subfield form a group of transformations which map each root to another or the same root. Galois used this group of symmetries to show which polynomials have roots which can be expressed as algebraic functions of the coefficients. The Norwegian mathematician Sophus Lie in his 1871 doctoral thesis replaced polynomial equations with differential equations and roots with solutions and so began the use of group theoretic methods to describe the solutions of ordinary and partial de's. In the 20th Century, all sorts of mathematical structures have been studied by means of their groups of symmetries. The work in this department of Cheryl Praeger, Alice Niemeyer and Cai Heng Li on graphs is an example, not to mention my own research on modules (structures like vector spaces, except that the scalars can be elements of any ring, not necessarily a field). But it is not only in mathematics that symmetries are studied by means of groups. To begin with, the word "symmetry" comes from aesthetics and symmetries are used and studied in art and architecture. In Chemistry, the symmetries of crystal structures are studied in X-ray crystallography to investigate the arrangement of atoms and so identify chemical compounds. Even in analytic chemistry, 3-dimensional symmetry is used to classify stereo-isomers, i.e. different molecules with the same chemical formula. In Physics group theory and symmetries are used in relativity theory, quantum theory and to describe and even predict fundamental particles and their properties.
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