of Symmetry in Physics,
Chemistry, and Mathematics
This page contains many links to the
History of Mathematics archive at the University of St Andrews, and to the
More about symmetries from this server
An introduction to elementary particle theory [Swedish].
Symmetries in physics
Links to related Web resources
This page is still under construction. Corrections and contributions
- 400 Description of the
5 Platonic solids.
- 300 The geometry of
polyhedra described by
1528 'De symmetria partium' by
Albrecht Dürer, a study of symmetry in art.
1596 In 'Mysterium Cosmographicum'
Johannes Kepler suggests that the orbits of the then known planets are defined
by the Platonic solids.
1609 Kepler publishes 'Astronomia Nova' where he announces his
three famous laws for planetary motion. The second law we can now understand
as the conservation of angular momentum, a consequence of the O(3) symmetry
of the gravitational force from the sun.
1611 In 'De nive sexangula' Kepler studies the hexagonal symmetry of
1669 Investigation of crystal angles published in 'De solido intra
sodium naturaliter contendo' by
Steensen, the Danish cleric, geologer and anatomist. Probably the first instance of
the 'law of constancy of angles', here for quartz crystals.
1687 'Principia' by
Isaac Newton, where the first law states the conservation of momentum due to the
homogeneity of space (translation invariance).
1770 Permutations first studied by
Joseph-Louis Lagrange in a paper on algebraic equations.
Jean-Babtiste Romé De Lisle publishes 'Essai de Cristallographie'.
He confirmed the observations of Steno, and later tried to
order crystals into symmetry classes.
René-Just Haüy publishes 'Essai d'une theorie sur la structure des cristaux'
describing experiments on the cleaving of crystals. Proposed that a
crystalline solid consists of replicas of a unit cell, and the
'law of rational indices'.
1830 J.F.C. Hessel derives the 32 crystal classes, starting from
the law of rational indices.
Evariste Galois is the first to understand the relation between the algebraic
solutions of an equation and the structure of a group of permutations
associated with the equation. This work was not published until 1846.
Cauchy studies the group properties of permutations. The permutations
of a fixed number of N elements is now called the symmetric group
Auguste Bravais derives the 14 space lattices in 3 dimensions.
1860 Louis Pasteur
discovers the connection between optical activity and enantiomorphic
molecular structures. Chiral molecules which are mirror images rotate light in
Felix Klein proposes the Erlangen program where geometry is
classified by invariance groups.
Arthur Cayley formulates the abstract group concept.
Derivation of the 230 space groups in 3 dimensions (independently) by
A.M. Schönflies and E.S. Fedorov
Sophus Lie and
publish 'Theorie der Transformationsgruppen'.
FitzGerald suggests what is later called the FitzGerald-Lorentz contraction,
introduce the transformations which make up what is now called the Lorentz group.
It is shown that they leave Maxwell's equations invariant. The Lorentz group with the space-time
translations added is often called the Poincaré group.
In his most famous paper
Einstein gives a set of physical assumptions from which the Lorentz transformations follow.
He thus creates
Special Relativity as a physical theory and an alternative to the Newtonian theory.
The latter uses a different set of transformations connecting the inertial reference frames,
namely the Galilei group.
Schur create the theory of group representations.
1912 Experimental evidence for the lattice structure of crystals
through x-ray diffraction by Friedrich and Knipping, following a
Emmy Noether shows the general connection between symmetries and
Hermann Weyl introduces a classical unified field theory for gravitation and
electromagnetism. It includes invariance under scale transformations, called
gauge invariance, which implies the conservation of electric charge.
S.N. Bose introduces what is now called Bose-Einstein statistics for
photons. In 1925 there is a generalization, by Einstein, to those particles or quanta
we now call
Their many-quanta states are invariant under all permutations.
Wolfgang Pauli proposes the 'exclusion principle', later called the 'Pauli
principle' for the states of electrons.
Werner Heisenberg and Pascual Jordan introduce the quantum theory of angular momentum and
Fermi-Dirac statistics introduced (by Fermi
!) for those
particles (e.g. electrons) we now call
Their many-particle states change sign under odd permutations.
This statement includes the Pauli principle.
Fritz London and Weyl introduce gauge transformations into quantum theory,
with total electric charge as the conserved quantity.
Dirac proposes a relativistic wave equation for spin 1/2 particles,
i.e. one covariant under the Poincaré group.
Weyl publishes 'Gruppentheorie und Quantenmechanik'.
Felix Bloch describes the electron wave functions in periodic potentials.
derives the splitting of atomic levels resulting
from the crystal field symmetry.
Eugene Wigner studies the effects of the symmetry of molecular configurations on the
introduces time reversal symmetry (T)
into quantum theory and publishes 'Gruppentheorie und ihre Anwendung auf der
Quantenmechanik der Atomspektren'.
Pauling studies the theory of chemical bonding using the symmetries of
orbitals. He receives the
Nobel Prize in Chemistry in 1954.
Heisenberg introduces a symmetry between protons and neutrons
in nuclear theory, it is later called isospin symmetry.
finds the positron in a cosmic ray experiment, the first of
the antiparticles (predicted by Dirac in 1931).
B.L van der Waerden: 'Die gruppentheoretische Methode in der
1935 V. Fock derives the spectrum of the H-atom from the SO(4)
Heisenberg introduces charge conjugation (C) as a symmetry operation
connecting particle and antiparticle states.
Frederick Seitz works out the representation theory of space groups, the symmetry
groups of crystal lattices.
1937 H.A. Jahn and E. Teller derive a connection between the symmetry of
molecular configurations and the stability of degenerate molecular electron
orbitals (Jahn-Teller effect): for a non-linear molecule there is always a
distortion into a shape of lower symmetry to remove any orbital degeneracy
of its electronic state.
1939 Wigner studies the unitary representations of the Poincaré
group. The results allow us to classify all relativistic wave equations and
the transformation properties of quantum fields.
1940 Pauli proves the spin-statistics theorem: particles with
half-integer spin have Fermi-Dirac statistics, those with integer spin are
H.S.M. Coxeter publishes 'Regular Polytopes'. Coxeter groups are
groups of reflections acting on such polytopes. They are important
in the theory of Lie algebras.
Yang and Mills introduce local isospin transformations as an internal
symmetry, i.e. they are transformations of the field operators which
depend on the point in space-time.
Wick, Wightman and Wigner introduce the notion of superselection rule.
1954-5 The PCT theorem is proved by Lüders and Pauli, involving
space inversion (P), charge conjugation (C) and time reversal (T): in a
local quantum field theory the product PCT of these transformations is always a
A parity breaking weak interaction is proposed by C.N. Yang and T-D. Lee
and verified experimentally by C.S. Wu. Yang and Lee share the 1957 Nobel prize
1959-61 Heisenberg, Goldstone and Nambu suggest that the ground state (vacuum)
of relativistic quantum field theory may lack the full global
symmetry of the Hamiltonian, and that massless excitations (Goldstone bosons)
must accompany this 'spontaneous symmetry breaking'.
In 1964 Higgs and others find that for spontaneously broken gauge symmetries there
are no Goldstone bosons but instead massive vector mesons (Higgs phenomenon).
1961 Murray Gell-Mann
and Yuval Neeman propose SU(3) as a symmetry for the strong
interactions (the Eightfold Way). This includes the isospin symmetry in a larger
symmetry group which also acts on the strangeness quantum number.
In 1964 Gell-Mann and Zweig propose a new, deeper, level of quanta, the quarks,
to account for the SU(3) symmetry.
1964 The CP breaking part of the weak interaction is found experimentally
J.W. Cronin and W.L. Fitch.
describe how the conservation of orbital symmetry
influences the course of molecular reactions, the 'Woodward-Hoffman rules'.
The essential features of the presently accepted
of particle physics are established.
1977 Roger Penrose demonstrates an
aperiodic tiling of the plane using only two different tiles
and an approximate 5-fold symmetry.
- 1984 D. Shechtman finds the first
quasicrystal in the laboratory
with evidence of dodecahedral structure, one not expected to exist by
Curl, Kroto, Smalley
and coworkers produce the first observed
molecules by laser-vaporizing graphite in a jet of helium. This discovery
was awarded the Nobel prize in Chemistry 1996.
Jan 22, 2005