showed that aperiodic structures transform into higher order structures (crystal lattices) or other quasicrystalline solids [@steurer2; @tsai; @koster]. These transitions are characterised by the occurrence of a symmetry breaking, thus allowing for their analysis in the framework of the phenomenological Landau theory for phase transitions [@sethna; @landau]. Similar transformations occur in virology; specifically, viruses undergo conformational changes as part of their maturation process, important for becoming infective, resulting in an expansion of the capsid [@erav]. Experiments and theoretical analysis [@zappa] proved that the initial and final states of the transformation retain icosahedral symmetry; however, little is known about the possible transition paths, which according to a first mathematical approach are likely to be asymmetrical [@paolo2].

In this paper we characterise structural transitions of icosahedral quasicrystals and point arrays in a group theoretical framework. The starting point is the crystallographic embedding of the icosahedral group into the point group of the hypercubic lattices in $\mathbb{R}^6$ [@levitov; @zappa1; @senechal], which is a standard way of defining icosahedral quasicrystals with the cut-and-project method. We then construct continuous transformations between icosahedral structures based on the Schur rotation method, originally introduced by Kramer *et al.* [@kramer; @kramer2], for the case of transitions between cubic and 