Related papers: The $E_8$ geometry from a Clifford perspective
Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators but their symmetry…
Introducing products between multivectors of Cl(0,7) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the…
We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4)…
We present a geometric construction of the exceptional Lie algebras F4 and E8 starting from the round 8- and 15-spheres, respectively, inspired by the construction of the Killing superalgebra of a supersymmetric supergravity background.…
Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing,…
Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…
Division algebras have demonstrated their utility in studying non-associative algebras and their connection to the Standard Model through complex Clifford algebras. This article focuses on exploring the connection between these complex…
Deformed $\mathfrak{g}_2$ exceptional applications are introduced via the Clifford algebra-parametrized formalism. Using the products between multivectors of $\cl_{0,7}$, the Clifford algebra over the metric vector space $\RR^{0,7}$, and…
Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad…
Vertices of the 4-dimensional semi-regular polytope, the \textit{grand antiprism} and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the \textbf{$E_{8}…
This paper investigates several distinct attempts to generalize in higher dimension the standard 2-dimensional phyllotaxy set construction. We first recall known contructions for these sets on $2D$ manifolds of constant curvature (the…
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…
4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group W(H4) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary…
How can we understand the origins of highly symmetrical objects? One way is to characterize them as the solutions of natural optimization problems from discrete geometry or physics. In this paper, we explore how to prove that exceptional…
We develop a systematic algorithm to construct, classify and study exact solutions of type II A/B supergravity which are time--dependent and homogeneous and hence represent candidate cosmological backgrounds. Using the formalism of solvable…
A generalization of the term "generalized Clifford algebras" (as appears in papers on advances in applied Clifford algebras) is introduced. This algebra is studied by means of structure theory of central simple algebras. A graph theoretical…
We examine certain maps from root systems to vector spaces over finite fields. By choosing appropriate bases, the images of these maps can turn out to have nice combinatorial properties, which reflect the structure of the underlying root…
We study the geometry of elliptic fibrations given by Weierstrass models resulting from Step 6 of Tate's algorithm. Such elliptic fibrations have a discriminant locus containing an irreducible component $S$, over which the generic fiber is…
Starting from basic identities of the group E8, we perform progressive reductions, namely decompositions with respect to the maximal and symmetric embeddings of E7xSU(2) and then of E6xU(1). This procedure provides a systematic approach to…
Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B…