I have now constructed E8 and in fact all other exceptional geometries (which are in 4D) within the 3D space we inhabit. This construction is very intuitive and does not require much advanced mathematics, but allows us to explain why these beautiful higherdimensional structures exist and exhibit the strange symmetries that they have, within the geometry of three dimensions that we are so familiar with. In particular, E8 can be constructed from the icosahedron (twentysided dice, one of the five Platonic solids), which is the shape that most viruses have for reasons of genetic economy, or the symmetry of footballs. This new view of the exceptional geometries as 3D phenomena in disguise has the potential to open large areas of mathematics and physics up for reinterpretation in this new way as well as a wealth of new insights.
The threedimensional world we know not only has three orthogonal directions (height, width, depth) but in fact also three orthogonal planes (e.g. the floor and two walls at right angles), a volume and a number (a point), which when taken together behave like an eightdimensional space. In this Clifford algebra approach, socalled spinors live in a 4D subset of the eight dimensions (the point and the three planes) with pairs of them describing the same rotation in 3D, whilst pairs of the objects in the whole 8D space describe the reflections. The icosahedron has 120 reflection and rotation symmetries, which are therefore doubly covered by 240 Clifford elements in 8D  these can be shown to in fact be E8, unveiling that it is actually a 3D phenomenon in disguise. Similarly, the exceptional 4D symmetries are generated by the spinors associated with the symmetries of all the Platonic solids. Particle physicists have a good understanding of these structures as Lie groups or Lie algebras, but ultimately they are defined by their reflection symmetries, and furthermore the root systems that in turn generate those reflection symmetries, so that these concepts are mostly used interchangeably. In our work on the symmetries of viruses, we have shown root systems to be a useful concept, in particular that of icosahedral symmetry  this, however, does not have a Lie group or algebra associated with it, so is not usually considered in particle physics. Thus this unique combination of working with the Platonic root systems in mathematical virology on the one hand, and the unusual Clifford algebraic approach on the other, has allowed this fundamental new insight. Usually when one argues for higherdimensional theories one considers them as fundamental, and we might only experience a part of this whole structure in our 3D world. These new results completely subvert this point of view by showing that these `obscure' higherdimensional symmetries actually have `space' to fit into the 3D geometry of our natural world. 
Since the dawn of time, human beings have striven to unravel the laws that govern the physical world around us. In our own time, String Theory and Grand Unified Theories are our most prominent contenders towards a Theory of Everything. These theories require higher dimensions or higherdimensional symmetries, e.g. superstrings live in ten dimensions  for good reasons, though this might seem esoteric and very different from the world we actually experience. In particular, a symmetry called E8 in eight dimensions is very interesting in mathematics, as it is the largest symmetry that does not have counterparts in every dimension, e.g. a form of `cube' exists in any dimension but E8 is `exceptional' in that way, and surprisingly it also features very prominently in String theory and Grand Unified Theories.

Root systems and Coxeter Groups
A reflection group paradigm

Clifford algebras
A good way to do geometry  also very good at reflections!
