## Coxeter Groups

Coxeter groups are abstract groups describable in terms of mirror symmetries. The finite Coxeter groups correspond to the finite Euclidean reflection groups. They include the symmetry groups of the Platonic solids as well as the Weyl groups of simple Lie algebras.

Finite Coxeter groups describe the properties of natural structures, e.g. of a viral protein container or a carbon onion, at a given radial level. In order to obtain information on how structural properties at different radial levels are collectively constrained by symmetry, affine extensions of these groups need to be considered. Such affine extended, infinite Coxeter groups can be constructed from the finite ones by introducing affine reflections, i.e. reflection planes not passing through the origin. Examples of such affine extensions of finite Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras. While infinite counterparts to the crystallographic finite Coxeter groups have been intensively studied, much less is known for the non-crystallographic equivalents; whilst crystalline arrangements and their crystallographic reflection symmetries are undoubtedly very important, the largest reflection groups in two and three dimensions are actually non-crystallographic: the symmetries of the regular polygons in two dimensions I2(n), and the symmetry group of the icosahedron in three dimensions H3=I_h. For example, in many cases the proteins in viral capsids are organised according to (rotational) icosahedral symmetry I. Thus symmetry is an important principle for virus structure, assembly and dynamics. In particular, this work revealed a previously unappreciated molecular scaling principle in virology, relating the structure of the viral capsid of Pariacoto virus to that of its packaged genome. This suggests that the overall organisation of such viruses follows an affine version of the icosahedral group, and implications of this discovery for virus dynamics and assembly have been discussed based on this new principle.

Finite Coxeter groups describe the properties of natural structures, e.g. of a viral protein container or a carbon onion, at a given radial level. In order to obtain information on how structural properties at different radial levels are collectively constrained by symmetry, affine extensions of these groups need to be considered. Such affine extended, infinite Coxeter groups can be constructed from the finite ones by introducing affine reflections, i.e. reflection planes not passing through the origin. Examples of such affine extensions of finite Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras. While infinite counterparts to the crystallographic finite Coxeter groups have been intensively studied, much less is known for the non-crystallographic equivalents; whilst crystalline arrangements and their crystallographic reflection symmetries are undoubtedly very important, the largest reflection groups in two and three dimensions are actually non-crystallographic: the symmetries of the regular polygons in two dimensions I2(n), and the symmetry group of the icosahedron in three dimensions H3=I_h. For example, in many cases the proteins in viral capsids are organised according to (rotational) icosahedral symmetry I. Thus symmetry is an important principle for virus structure, assembly and dynamics. In particular, this work revealed a previously unappreciated molecular scaling principle in virology, relating the structure of the viral capsid of Pariacoto virus to that of its packaged genome. This suggests that the overall organisation of such viruses follows an affine version of the icosahedral group, and implications of this discovery for virus dynamics and assembly have been discussed based on this new principle.

The principle of affinisation, i.e. the extension of a finite symmetry group by the addition of a non-compact generator, is commonly used in the context of crystallographic groups to generate space groups. It has been introduced in a non-crystallographic setting for the first time by Twarock and Patera. In particular, in this reference the reflection groups H_3 and H_4, which are the only reflection groups containing icosahedral symmetry as a subgroup, have been extended by an affine reflection, and it has been shown that in combination with generators of the finite groups the affine reflections act as translations. Subsequently, we have classified affinisations of icosahedral symmetry via translation operators. They contain the affinisations derived previously, but in addition provide a much wider spectrum of extended group structures.

We illustrate the construction principle geometrically here for the two-dimensional example of the rotational symmetry group C_5 of a regular pentagon. Addition of the translation operator T (here taken to be of the same length as the golden ratio $\tau$ times the radius of the circle into which the pentagon is inscribed) makes the group non-compact by creating a displaced version of the original pentagon. The action of the symmetry group C_5 of the pentagon generates additional copies in such a way that, after removal of all edges, a point array is obtained that has the same rotational symmetries as the original pentagon. Since every point in the array is related to every other via application of generators of the extended group, all points can be generated from a single point via the action of the extended group. They are hence collectively encoded by the group structure.

The points in the array correspond to words in the generators of the affine extended group. Thus, if points are located in more than one of the translated and rotated copies of the original pentagon, then these points, called coinciding points, correspond to non-trivial relations between group elements, and the extended group is hence not the free group. The point set obtained here with the translation of length τ=(1+√5)/2 has cardinality 25, as opposed to 30, which would be the value in the generic case. Translations giving rise to such coinciding points are hence distinguished from a group theoretical point of view.

Note that the point array contains a composition of a pentagon and a decagon of different scaling, both centred on the origin. The affine group determines their relative sizes (or radial levels), and hence introduces radial information in addition to that encoded by the original group structure. Affine symmetry therefore allows one to constrain the overall geometry of a multishell structure from just part of the blueprint, c.f. the applications to virus structure and to carbon onions.

We illustrate the construction principle geometrically here for the two-dimensional example of the rotational symmetry group C_5 of a regular pentagon. Addition of the translation operator T (here taken to be of the same length as the golden ratio $\tau$ times the radius of the circle into which the pentagon is inscribed) makes the group non-compact by creating a displaced version of the original pentagon. The action of the symmetry group C_5 of the pentagon generates additional copies in such a way that, after removal of all edges, a point array is obtained that has the same rotational symmetries as the original pentagon. Since every point in the array is related to every other via application of generators of the extended group, all points can be generated from a single point via the action of the extended group. They are hence collectively encoded by the group structure.

The points in the array correspond to words in the generators of the affine extended group. Thus, if points are located in more than one of the translated and rotated copies of the original pentagon, then these points, called coinciding points, correspond to non-trivial relations between group elements, and the extended group is hence not the free group. The point set obtained here with the translation of length τ=(1+√5)/2 has cardinality 25, as opposed to 30, which would be the value in the generic case. Translations giving rise to such coinciding points are hence distinguished from a group theoretical point of view.

Note that the point array contains a composition of a pentagon and a decagon of different scaling, both centred on the origin. The affine group determines their relative sizes (or radial levels), and hence introduces radial information in addition to that encoded by the original group structure. Affine symmetry therefore allows one to constrain the overall geometry of a multishell structure from just part of the blueprint, c.f. the applications to virus structure and to carbon onions.