The potency of pattern: Mind the gap

That’s Maths: Symmetry is invariance under transformation and it is described by a mathematical discipline — group theory

In his book A Mathematician’s Apology, leading British mathematician G H Hardy wrote: “A mathematician, like a painter or poet, is a maker of patterns.” The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; beauty is the acid test, he observed.

Everyone is familiar with the concept of symmetry, although most of us would struggle to define it. Beauty and symmetry are inextricably linked. For a mathematician, symmetry is invariance under transformation, and it is described by the mathematical discipline called group theory.

The foundations of group theory were laid down in 1832 by Évariste Galois, who perished in a stupid duel before his 21st birthday. He studied discrete groups, with a finite number of elements. The Norwegian mathematician Sophus Lie extended this theory to include continuous groups.

The symmetries of polygons are represented by finite groups; the infinite symmetries of a circle are modelled by continuous Lie groups. These groups, which include the group SU(3) mentioned below, are now essential tools in theoretical physics.


Revealing Gaps

In 1869, Dmitri Mendeleev proposed a brilliant arrangement of the chemical elements according to the magnitude of atomic weights, that revealed periodic changes of their properties. He achieved the best pattern — the maximum symmetry — by moving some elements and by leaving some gaps in his table, boldly predicting that they would be filled by elements yet to be discovered.

He anticipated the physical and chemical properties of these elements. Many chemists were sceptical but, within a few years, gallium was discovered, fitting well with Mendeleev’s prophecy. Soon afterwards, the remaining gaps were filled. The periodic table has been of inestimable value in chemistry.

About 1950, physicists thought they had complete knowledge of the elements of matter: protons, neutrons, electrons and photons. Soon, many new particles were discovered, leading to the chaotic era of the particle zoo. In 1961, physicist Murray Gell-Mann found that families of subatomic particles called hadrons could be arranged systematically, based on their symmetry properties, in a regular pattern of octets that he called the eightfold way. An Israeli physicist, Yuval Ne’eman, proposed a similar model around the same time.

Gell-Mann noticed a particular gap in his pattern and predicted that experimentalists would soon find a new particle to fill it. He dubbed this particle the omega minus and, based on symmetry arguments, predicted its charge and mass. Three years later, a particle fitting this description was discovered, confirming the essential value of Gell-Mann’s ideas.

Mendeleev could not explain the many patterns in his periodic table of elements. The relationships found by Gell-Mann between particles matched the mathematical symmetry group called SU(3). The octets in the eightfold way are representations of this group. Hadrons are not fundamental but are combinations of quarks held together by gluons.

The branch of physics dealing with these is called quantum chromodynamics and it greatly elucidates the patterns found in Mendeleev’s table. For his classification of subatomic particles, Gell-Mann was awarded the Nobel Prize in Physics in 1969, just 100 years after Mendeleev devised his periodic table.

  • Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at