April 12, 2022
In this paper we discuss reflection groups and root systems, in particular non-crystallographic ones, and a Clifford algebra framework for both these concepts. A review of historical as well as more recent work on viral capsid symmetries motivates the focus on the icosahedral root system $H_3$. We discuss a notion of affine extension for non-crystallographic groups with applications to fullerenes and viruses. The icosahedrally ordered component of the nucleic acid within the virus capsid and the interaction between the two have shed light on the viral assembly process with interesting applications to antiviral therapies, drug delivery and nanotechnology. The Clifford algebra framework is very natural, as it uses precisely the structure that is already implicit in this root system and reflection group context, i.e. a vector space with an inner product. In addition, it affords a uniquely simple reflection formula, a double cover of group transformations, and more insight into the geometry, e.g. the geometry of the Coxeter plane. This approach made possible a range of root system induction proofs, such as the constructions of $E_8$ from $H_3$ and the exceptional 4D root systems from 3D root systems. This makes explicit various connections between what Arnold called Trinities (sets of three exceptional cases). In fact this generalises further since the induction construction contains additional cases, namely two infinite families of cases. It therefore actually yields $ADE$ correspondences between three sets of different mathematical concepts that are usually thought of separately as polytopes, subgroups of $SU(2)$ and $ADE$ Lie algebras. Here we connect them explicitly and in a unified way by thinking of them as three different sets of root systems.
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July 20, 2012
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) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant ch...
December 6, 2018
In this paper we present novel $ADE$ correspondences by combining an earlier induction theorem of ours with one of Arnold's observations concerning Trinities, and the McKay correspondence. We first extend Arnold's indirect link between the Trinity of symmetries of the Platonic solids $(A_3, B_3, H_3)$ and the Trinity of exceptional 4D root systems $(D_4, F_4, H_4)$ to an explicit Clifford algebraic construction linking the two ADE sets of root systems $(I_2(n), A_1\times I_2(...
February 18, 2016
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 metric that exists on these spaces can always be used to construct the corresponding Clifford algebra. Via the Cartan-Dieudonn\'e theorem all the transformations of interest can be written as products of reflections and thus via `sandwiching' with Clifford algebra multivectors. These mult...
February 18, 2016
This paper considers the geometry of $E_8$ from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system $H_3$ gives rise to the largest (and therefore exceptional) non-crystallographic root system $H_4$. Arnold's ...
February 18, 2016
We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description of reflections via `sandwiching'. This extends to a description of orthogonal transformations in general by means of `sandwiching' with Clifford algebra multivectors, since all orthogonal transformations can be written as products of reflect...
May 7, 2012
Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. ...
July 31, 2012
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 reflections yields rotations that in a Clifford algebra framework are described by spinors. In three dimensions these spinors themselves have a natural four-dimensional Euclidean structure, and discrete spinor groups can therefore be interpreted as 4D polytopes. In fact, we show that these polyt...
February 18, 2016
$E_8$ is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional space very different from the space we inhabit; for instance the Lie group $E_8$ features heavily in ten-dimensional superstring theory. Contrary to that point of view, here we show that the $E_8$ root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of three-dimensional geometry. The $240$ roo...
October 24, 2011
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 systems E_8, D_6 and A_4. We show that the induced affine extensions of the non-crystallographic groups H_4, H_3 and H_2 correspond to a distinguished subset of the Kac-Moody-type extensions considered in Dechant et al. This class of extensions was motivated by physical applications in i...
March 13, 2021
Recent work has shown that every 3D root system allows the construction of a correponding 4D root system via an `induction theorem'. In this paper, we look at the icosahedral case of $H_3\rightarrow H_4$ in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonn\'e theorem, giving a simple construction of the Pin and Spin covers. Using this connection with $H_3$ via th...