From symmetry to particles

The notion of a particle-like state emerging from a symmetry breaking is given five corresponding pictures. We start from a geometrical picture in two dimensions involving a modular curve constructed using 336 triangles. The same number of building blocks is found again, this time as 336 contact points in the ten dimensional space of super string theory in the context of the largest kissing number of lattice sphere packing. The next corresponding representation is an abstract one pertinent to the order of the simple linear Lie group SL(2, n) in seven dimensions (n = 7) which leads to 336 symmetries. Subsequently a tensorial picture is given using the Riemannian tensor of relativity theory but this time in an eight dimensional space (n = 8) for which the number of independent components is again 336. Finally we use a physical string theory related picture in the 12 dimensions of F theory to find 336 moduli space dimensions representing the instanton cells of our theory. It is evident that the five preceding pictures are ten fold interconnected and exchangeable. This additional mental freedom does not only enhance the feeling of understanding, but also facilitates the easy recognition of complex mathematical relations and its connection to the physical concepts.