A Very Intuitive Geometric Picture of the 24-cell, E8 and Λ16 Lattices Given by Using the Hopf Maps

We use the Hopf fibrillation to give simple and intuitive geometric constructions of the 24-cell, E8 and Λ16 lattices. The n-dimensional kissing problem asks how many non overlapping unitn-dimensional spheres can be placed touching a central unit n-dimensional sphere? This question has applications to making efficient codes and other problems [1]. The two dimensional sphere is the disk of points x + y = 1, and by using N identical circular coins the reader can easily convince oneself that the kissing number in two dimensions (K2) is six. Similarly, the one dimensional sphere is a line segment, and K1 = 2. Already in three dimensions the problem becomes much more interesting. Indeed, the Seventeenth century featured a dispute between Isaac Newton who believed that K3 = 12, and David Gregory who thought that K3 = 13. Perhaps not surprisingly, Newton was correct but it took more than two and one half centuries to prove this[9]. Using clever linear programming arguments it was proven some decades ago that K8 = 240 and K24 = 196560 [4, 7] and that K4 = 24 or 25. Recently, a proof that K24 = 24 using much more extensive and subtle use of linear programming has be presented[6], however, the proof has not yet been fully vetted [8]. There exists a map known as the (first) Hopf map between the surface of a three dimensional sphere and the surface of a four dimensional sphere, which essentially constructs the surface of a four dimensional sphere by considering there to be a circle of points at every point on the surface of a three dimensional sphere. Here we emphasize that the Hopf map also gives a very intuitive way of appreciating the N = 24 kissing configuration for S–known as the 24-cell and then use the second and third Hopf maps to give intuitive descriptions