# A Timeline of Symmetry in Physics,
Chemistry, and Mathematics.

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This page contains many links to the
MacTutor
History of Mathematics archive at the University of St Andrews.

Links to other interesting archives can be found at the bottom of the page.
The page is still under construction. Corrections and contributions
are invited.

**- 400** Description of the
5 Platonic solids.

**- 300** The geometry of
polyhedra described by
Euclid.

**1528** 'De symmetria partium' by
Dürer, a study of symmetry in art.

**1596** In 'Mysterium Cosmographicum'
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
hree famous laws for planetary motion. The 2nd 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
snow crystals.

**1669** Investigation of
crystal angles by Steno, alias Niklas Stensen,
the Danish geologer and anatomist.

**1687** 'Principia' by
Newton, where the first law states the conservation of momentum due to the
homogeneity of space (translation invariance).

**1830** Hessel derives the 32 crystal classes.

**1832**
Galois introduces the group concept and applies
it to permutations of the roots of equations. This work was not published
until 1846.

**1844**
Cauchy studies the theory of permutations. The permutations of a fixed
number of N elements is now called the symmetric group S_{N}.

**1849** Bravais derives the 14 space lattices in 3 dimensions.

**1860** Pasteur discovers the connection between optical activity and enantiomorphic
molecular structures. Chiral molecules which are mirror images rotate light in
opposite senses.

**1872**
Felix Klein proposes the Erlanger program where geometry is
classified by invariance groups.

**1878**
Cayley formulates the abstract group concept.

**1890-91**
Derivation of the 230 space groups in 3 dimensions by
Schönflies and Fedorov

**1893**
Lie and Engel: 'Theorie der Transformationsgruppen'.

**1886-1904**
FitzGerald suggests what is later called the FitzGerald-Lorentz contraction,
Larmor,
Lorentz and
Poincaré
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.

**1905**
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.

**1895-1910**
Frobenius and
Schur create the theory of group representations.

**1912** Experimental evidence for the lattice structure of crystals
through x-ray diffraction by Max von Laue and others.

**1918**
Emmy Noether shows the general connection between symmetries and
conserved quantities.

**1918**
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.

**1924**
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
bosons.
Their many-quanta states are invariant under all permutations.

**1925**
Pauli proposes the 'exclusion principle', later called the 'Pauli
principle' for the states of electrons.

**1926**
Born,
Heisenberg and Jordan introduce the quantum theory of angular momentum and
spin 1/2.

**1926**
Fermi-Dirac statistics introduced (by Fermi and Dirac!) for those
particles (e.g. electrons) we now call
fermions.
Their many-particle states change sign under odd permutations.
This statement includes the Pauli principle.

**1927-28**
London and Weyl introduce gauge transformations into quantum theory,
with total electric charge as the conserved quantity.

**1928**
Dirac proposes a relativistic wave equation for spin 1/2 particles,
i.e. one covariant under the Poincaré group.

**1928**
Weyl: 'Gruppentheorie und Quantenmechanik'.

**1929** Bloch describes the electron wave functions in periodic
potentials.

**1929**
Bethe derives the splitting of atomic levels resulting
from the crystal field symmetry.

**1930**
Wigner studies the effects of the symmetry of molecular configurations on the
vibrational spectrum.

**1931** Wigner introduces time reversal symmetry (T)
into quantum theory and publishes 'Gruppentheorie und ihre Anwendung auf der
Quantenmechanik der Atomspektren'.

**1931**
Pauling
studies the
theory of chemical bonding using the symmetries of orbitals.

**1932** Heisenberg introduces a symmetry between protons and neutrons in nuclear
theory, it is later called isospin symmetry.

**1932**
Carl Anderson finds the positron in a cosmic ray experiment, the first of
the antiparticles (predicted by Dirac in 1931).

**1932**
van der Waerden: 'Die gruppentheoretische Methode in der
Quantenmechanik'.

**1935** V. Fock derives the spectrum of the H-atom from the SO(4)
symmetry.

**1936**
Heisenberg introduces charge conjugation (C) as a symmetry operation connecting particle
and antiparticle states.

**1937** Jahn and 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
Bosons.

**1954**
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.

**1954**
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
symmetry.

**1956-7**
A parity breaking weak interaction is proposed by Yang and Lee and
verified experimentally by Wu. Yang and Lee share the 1957 Nobel prize
for physics.

**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** Gell-Mann and 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
by Cronin and Fitch.

**1965** Woodward and Hoffman describe how the conservation of orbital symmetry
influences the course of molecular reactions, the 'Woodward-Hoffman rules'.

**1973-74**
The essential features of the presently accepted
Standard Model
of particle physics are established.

**1977** Penrose demonstrates an
aperiodic tiling of the plane using only two different tiles
and an approximate 5-fold symmetry. In 1984 Shechtman et al
find the first quasicrystal in the laboratory with evidence of dodecahedral structure,
one not expected to exist by conventional wisdom.

**1985**
Curl, Smalley and coworkers produce the first observed
C60 molecules by laser-vaporizing graphite in a jet of helium.

**UP** -
Modified 16 Jan 2004