Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their duals the Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups $W(A_1 \oplus A_1 \oplus A_1)$, $W(A_3)$, $W(B_3)$ and $W(H_3)$ to derive the orbits representing the solids of interest. They lead to the pol...

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...

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...

In this paper, we take the classic dihedral and quaternion groups and explore questions like "what if we replace $i=e^{2\pi i/4}$ in $Q_8$ with a larger root of unity?" and "what if we add a reflection to $Q_8$?" The delightful answers reveal lesser-known families like the dicyclic, diquaternion, semidihedral, and semiabelian groups, which come to life with visuals such as Cayley graphs, cycle graphs, and subgroup lattices.

In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same time, Hestenes geometric algebra provides a didactically promising instrument to model phenomena in physics mathematically and in a tangible manner. Quaternions particularly have a catchy interpretation in the context of geometric algebra whi...

Vertices of the 4-dimensional semi-regular polytope, \textit{snub 24-cell} and its symmetry group $W(D_{4}):C_{3} $ of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of \textbf{$E_{8} $} root system. The quaternionic root system of $H_{4} $ splits as the vertices of 24-cell and the \textit{snub 24-cell} under the symmetry group of the \textit{snub 24-cell} which is one of the maximal subgroups of the group \textb...

This paper describes in detail how (discrete) quaternions - ie. the abstract structure of 3-D space - emerge from, first, the Void, and thence from primitive combinatorial structures, using only the exclusion and co-occurrence of otherwise unspecified events. We show how this computational view supplements and provides an interpretation for the mathematical structures, and derive quark structure. The build-up is emergently hierarchical, compatible with both quantum mechanics ...

Complementary idempotent paravectors and their ordered compositions, are used to represent multivector basis elements of geometric Clifford algebra for 3D Euclidean space as the states of a geometric byte in a given frame of reference. Two layers of information, available in real numbers, are distinguished. The first layer is a continuous one. It is used to identify spatial orientations of similar geometric objects in the same computational basis. The second layer is a binary...

This paper gives an explicit isomorphic mapping from the 240 real $\mathbb{R}^{8}$ roots of the $E_8$ Gosset $4_{21}$ 8-polytope to two golden ratio scaled copies of the 120 root $H_4$ 600-cell quaternion 4-polytope using a traceless 8$\times$8 rotation matrix $\mathbb{U}$ with palindromic characteristic polynomial coefficients and a unitary form $e^{\text {i$\mathbb{U}$}}$. It also shows the inverse map from a single $H_4$ 600-cell to $E_8$ using a 4D$\hookrightarrow$8D chir...

In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous paper. On the basis of new classification of Clifford algebra elements it is possible to reveal and prove a number of new properties of Clifford algebra. We use k-fold commutators and anticommutators. In this paper we consider Clifford and exterior degrees and elementary functions of Clifford algebra elements.