Quaternions and their use in treating the orientation of arbitrary rigid particles and their rotations are introduced. Explicit expressions needed to convert between quaternions and Euler angles representations are given.

This chapter starts with the equations of motion for atomistic systems and their time integration, including the multiple time step methods taking care of different timescales. Systems of rigid anisotropic particles are also discussed with the help of the quaternion formulation, avoiding spurious singularities. Constant temperature and constant pressure methods are considered. A summary of available molecular dynamics packages is provided.

Using quaternions, we give explicit formulas for the global symmetries of the three-dimensional Kepler problem. The regularizations of the Kepler problem that are based on the Hopf map and on stereographic projections, respectively, are interpreted in terms of these symmetries.

The main purpose of this chapter is to discuss groups generated by reflexions, concentrating here on the finite and discrete infinite groups in euclidean spaces. While establishing notation and conventions, there are surveys of the algebraic and metrical properties of euclidean spaces, and a treatment of the main features of convex sets that are appealed to subsequently. The classification of the finite and discrete infinite reflexion groups in euclidean spaces is a core feature; the initial part of the treatment is novel. There is then a brief description of subgroup relationships among these groups. Certain angle-sum relations for polytopes and cones are employed to find the orders of the finite Coxeter groups by purely elementary geometric methods; these are established here for polyhedral sets in general. The lower-dimensional spaces are somewhat special. The finite rotation groups in three dimensions are classified, and are shown to be subgroups of reflexion groups. Finally, there is an introduction to quaternions, which provide an alternative approach to finite orthogonal groups in 4-dimensional space; these are needed to describe certain regular polyhedra in that space.

As in the case of the 3-dimensional regular apeirohedra described in the previous chapter, the mirror vector plays an important role in the classification of the 4-dimensional regular polyhedra. Thus the first task is to determine the possible mirror vectors of such polyhedra. The polyhedra with mirror vector (3,2,3) and their relatives under standard operations such as Petriality form a specially rich family. One particular family of these polyhedra is treated in detail, with a description of their realization domains. With the mirror vector (2,3,2), most of the standard operations lead to polyhedra in the same class. Though there is a close analogy between the infinite and finite cases, those with mirror vector (2,2,2) have symmetry groups that need not be related to reflexion groups; the treatment here employs quaternions. There are various connexions among these regular polyhedra, the most interesting being the way that the skewing operation takes certain polyhedra in class (3,2,3) into polyhedra of class (2,2,2).

Classical regular polytopes are those whose symmetry groups are generated by hyperplane reflexions; these were the subject of Coxeter’s monograph ‘Regular Polytopes’, which first appeared in 1948. This chapter classifies these polytopes, and describes relationships among them. A new condition is given here to be a classical regular polytope. First treated are the three sequences of polytopes – simplex, staurotope and cube – that occur in all dimensions. The 24-cell is looked at next, introducing the useful role played by quaternions in several places. Next are described the familiar icosahedron and dodecahedron. Then the regular 600-cell is constructed and its realization domain found. The dual 120-cell is then described, followed by finding the regular star-polytopes and their abstract automorphism groups. Then the discrete regular apeirotopes or honeycombs are classified. The regular compounds of polytopes are then described, including some that are relatively new. Throughout, the Petrie polygons of these regular polytopes are found by elementary means, that is, not by solving trigonometric equations. The realization domain of the 120-cell is left until last, because describing it draws on much of the earlier material in the chapter.

The quaternions of Hamilton show that it possible to generalise the complex numbers, but only at the expense of introducing unexpected behaviour. It is thus reasonable to extend no further than the complex numbers. However the quaternions provided the inspiration for vectors.