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Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups
Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E8, have been used extensively in the literature. The present paper analyses such Coxeter groups inExpand
The birth of E8 out of the spinors of the icosahedron
It is shown that the E8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3D geometry, which opens up a novel interpretation of these phenomena in Terms of spinorial geometry. Expand
Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups
Motivated by recent results in mathematical virology, we present novel asymmetric -integer-valued affine extensions of the non-crystallographic Coxeter groups H2, H3 and H4 derived in aExpand
Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction
In this paper, we show that via a novel construction every rank-3 root system induces a root system of rank 4. Via the Cartan-Dieudonn\'e theorem, an even number of successive Coxeter reflectionsExpand
Viruses and fullerenes--symmetry as a common thread?
The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affines symmetry approach. Expand
Affine extensions of non-crystallographic Coxeter groups induced by projection
In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systemsExpand
Platonic solids generate their four-dimensional analogues.
This paper shows how regular convex 4-polytopes can be constructed from three-dimensional considerations concerning the Platonic solids alone, explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. Expand
Clifford Algebra is the Natural Framework for Root Systems and Coxeter Groups. Group Theory: Coxeter, Conformal and Modular Groups
In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metricExpand
A Clifford Algebraic Framework for Coxeter Group Theoretic Computations
Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) CoxeterExpand
Models of viral capsid symmetry as a driver of discovery in virology and nanotechnology
Viruses are prominent examples of symmetry in biology. A better understanding of symmetry and symmetry breaking in virus structure via mathematical modelling opens up novel perspectives on howExpand