A Timeline of Symmetry in Physics, Chemistry, and Mathematics

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Symmetries in physics [Course information]

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• - 400 Description of the 5 Platonic solids.
• - 300 The geometry of polyhedra described by Euclid.
• 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 snow crystals.
• 1669 Investigation of crystal angles published in 'De solido intra sodium naturaliter contendo' by Nicolaus Steno, aka Niels 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.
• 1772 Jean-Babtiste Romé De Lisle publishes 'Essai de Cristallographie'. He confirmed the observations of Steno, and later tried to order crystals into symmetry classes.
• 1784 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.
• 1832 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.
• 1844 Cauchy studies the group properties of permutations. The permutations of a fixed number of N elements is now called the symmetric group S_N.
• 1849 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 opposite senses.
• 1872 Felix Klein proposes the Erlangen program where geometry is classified by invariance groups.
• 1878 Arthur Cayley formulates the abstract group concept.
• 1890-91 Derivation of the 230 space groups in 3 dimensions (independently) by A.M. Schönflies and E.S. Fedorov
• 1893 Sophus Lie and Friedrich Engel publish '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 Friedrich and Knipping, following a suggestion by von Laue.
• 1918 Emmy Noether shows the general connection between symmetries and conserved quantities.
• 1918 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.
• 1924 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 bosons. Their many-quanta states are invariant under all permutations.
• 1925 Wolfgang Pauli proposes the 'exclusion principle', later called the 'Pauli principle' for the states of electrons.
• 1926 Max Born, Werner Heisenberg and Pascual 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 Fritz 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 publishes 'Gruppentheorie und Quantenmechanik'.
• 1929 Felix Bloch describes the electron wave functions in periodic potentials.
• 1929 Hans Bethe derives the splitting of atomic levels resulting from the crystal field symmetry.
• 1930 Eugene 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. He receives the Nobel Prize in Chemistry in 1954.
• 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 B.L 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.
• 1936 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 Bosons.
• 1948 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.
• 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 C.N. Yang and T-D. Lee and verified experimentally by C.S. 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 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 by J.W. Cronin and W.L. Fitch.
• 1965 R.B. Woodward and R. 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 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 conventional wisdom.
• 1985 Curl, Kroto, Smalley and coworkers produce the first observed C60 molecules by laser-vaporizing graphite in a jet of helium. This discovery was awarded the Nobel prize in Chemistry 1996.

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  Jan 22, 2005