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 trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and $E_8$. Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the $120$ elements of the icosahedral group $H_3$ are doubly covered by $240$ 8-component objects, which endowed with a `reduced inner product' are exactly the $E_8$ root system. It was previously known that $E_8$ splits into $H_4$-invariant subspaces, and we discuss the folding construction relating the two pictures. This folding is a partial version of the one used for the construction of the Coxeter plane, so thirdly we discuss the geometry of the Coxeter plane in a Clifford algebra framework. We advocate the complete factorisation of the Coxeter versor in the Clifford algebra into exponentials of bivectors describing rotations in orthogonal planes with the rotation angle giving the correct exponents, which gives much more geometric insight than the usual approach of complexification and search for complex eigenvalues. In particular, we explicitly find these factorisations for the 2D, 3D and 4D root systems, $D_6$ as well as $E_8$, whose Coxeter versor factorises as $W=\exp(\frac{\pi}{30}B_C)\exp(\frac{11\pi}{30}B_2)\exp(\frac{7\pi}{30}B_3)\exp(\frac{13\pi}{30}B_4)$.
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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...
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. ...
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 ...
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...
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...
June 26, 2008
We consider a straightforward extension of the 4-dimensional spacetime $M_4$ to the space of extended events associated with strings/branes, corresponding to points, lines, areas, 3-volumes, and 4-volumes in $M_4$. All those objects can be elegantly represented by the Clifford numbers $X\equiv x^A \gamma_A \equiv x^{a_1 ...a_r} \gamma_{a_1 ...a_r}, r=0,1,2,3,4$. This leads to the concept of the so-called Clifford space ${\cal C}$, a 16-dimensional manifold whose tangent space...
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(...
November 2, 2023
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...
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...