The concept to the metric is introduced. Various geometries, both flat and curved, are described including Euclidean space; Minkowski space-time; spheres; hyperbolic planes and expanding space-times. Lorentz transformations and relativistic time dilation in flat space-time is discussed as well as gravitational red-shift and the Global Positioning System. Hubble expansion and the cosmological red-shift are also explained.

This preliminary chapter contains notations, definitions, and basic concepts needed for the study of Measure Theory and Functional Analysis. Most of this chapter is for reference and may be read only as needed. Included are concepts such as Convergence, Continuity, and Compactness in Euclidean Spaces. The theory of Euclidean Measure and sets of Measure Zero are covered. An overview is included of Integration sufficient to begin the study of Functional Analysis. The chapter finishes with topics such as Functions of Bounded Variation, Inequalities, along with a discussion of the Axiom of Choice.

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.

At the cost of some redundancy, to facilitate accessibility, the multiresolution decomposition and inversion of symmetric positive definite (SPD) matrices on finite-dimensional Euclidean space are developed in the Gamblet Transformand Decomposition framework.