Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups

In this paper, we show how regular convex 4-polytopes - the analogues of the Platonic solids in four dimensions - can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan-Dieudonne theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arisin...

These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of t...

This paper is to serve as a key to the projective (homogeneous) model developed by Charles Gunn (arXiv:1101.4542 [math.MG]). The goal is to explain the underlying concepts in a simple language and give plenty of examples. It is targeted to physicists and engineers and the emphasis is on explanation rather than rigorous proof. The projective model is based on projective geometry and Clifford algebra. It supplements and enhances vector and matrix algebras. It also subsumes comp...

We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra $\mathcal{C}\ell_{3,3}$ of the quadratic space $\mathbb{R}^{3,3}$. We show that this algebra describes in a unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using the concept of Hodge duality, we define an operation called cotranslation, and show that the operation of perspecti...

We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford's geometric algebra. Consequently, we establish a connection between a three dimensional icosahedral seed, a six dimensional Dirichlet quantized host and a higher dimensional lattice structure. The 20G, owing to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet integer re...

We define in an axiomatic fashion a \emph{Coxeter datum} for an arbitrary Coxeter group $W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear paring without any integrality and non-degeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root s...

In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to define Clifford algebras with scalars in arbitrary rings and provides new suggestions for an infinite-dimensional approach. Furthermore, I give a quick review of classic results regarding geometric algebras, such as their classification in...

Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural "Coxeter element". The folding of these graphs and groups is also discussed, using the theory of C-algebras. (Proceedings of the Taniguchi Symposium {Topological Field Theory, Primitive Forms and Related Topics}, Kyoto Dec 1996)

This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis of geometric algebras. Geometric algebras have been expanded to conformal geometric algebras, which provide an ideal framework for modern computer graphics. Within this framework a new visualization of three-dimensional crystallographic spa...

In this article we present a new and not fully employed geometric algebra model. With this model a generalization of the conformal model is achieved. We discuss the geometric objects that can be represented. Furthermore, we show that the Pin group of this geometric algebra corresponds to inversions with respect to axis aligned quadrics. We discuss the construction for the two- and three-dimensional case in detail and give the construction for arbitrary dimension. Key Words: C...