That fivefold symmetry should play a crucial role in the non-equilibrium behaviour of con- densed matter was proposed in the 1950s . Six decades later, the basic mechanism of the solidification of liquids remains unexplained, either in the case that the material crys- tallises, or that it forms an amorphous solid, a glass . We will explore the implications of fivefold symmetry in the solidification of liquids and discuss two recent developments.
Crystallisation is among the most common everyday physical phenomena. Yet in the only material in which quantitative comparison has been made between experiment and theory — hard spheres — predictions of crystal nucleation rates are up to 20 orders of magnitude slower than measurements, the “second worst prediction in physics” . This discrepancy casts doubt upon the theoretical methods concerned — importance sampling — which is important not only for crystallisation, because these methods are used to tackle a very wide range of problems, such as drug uptake in cells and chemical reaction pathways. We present results that show that fivefold symmetric arrangements of particles may hold the key to resolving this long-standing puzzle .
The nature of amorphous solids — glasses — is not understood: the possibility of a phase transition to a thermodynamically stable “ideal glass” is a contentious and challenging issue. Unlike everyday non-equilibrium glasses, such an ideal glass has a vanishing entropy — like a crystal — yet remains amorphous. Building on the ideas of Frank, the geometric frustration approach to the glass transition posits an avoided phase transition in a curved space inaccessible to experiment . Here we show that such a “crystallisation” to a state comprised of fivefold symmetric icosahedra indeed occurs and consider the implications of this avoided transition in the Euclidean space relevant to experiments .
 F.C. Frank Proc. R. Soc. A., 215 43 (1952)
 C.P. Royall and S.R. Williams, Phys. Rep., 560 1 (2015)
 J. Russo, A. Maggs, H. Tanaka and D. Bonn, Soft Matter, 9 7369 (2013)
 Taffs J and Royall CP, Nature Communications, 7 13225 (2016)
 G. Tarjus, S. A. Kivelson, Z. Nussinov, and P. Viot, J. Phys.:Condens. Matter 17 R1143 (2005)
 F. Turci, G. Tarjus and C.P. Royall, Phys. Rev. Lett. 118, 215501 (2017).