All possible transformations of the mathematical (Yang-Mills) fields whose spin-1 quanta mediate the forces between particles belong to a symmetry group. There are four types of such Lie groups, along with five 'exceptional groups'. The largest of the exceptional groups is the rank-8 Lie group called 'E8'. It is characterized by 248 roots, mathematically represented by 8-dimensional root vectors expressed as 8-tuples of the numbers 0, ±½ & ±1 (see diagram opposite). Eight are so-called 'zero roots,' or simple roots, and 240 are 'non-zero roots.' Associated with each root is a 10-dimensional, Yang-Mills gauge field. E8×E8 heterotic superstring theory predicts that 248 gauge bosons (spin-1 particles) mediate the interactions between superstrings of ordinary matter in 10-dimensional space-time. E6 is an exceptional subgroup of E8 that physicists have considered as an intermediate stage in the breakdown of the symmetry of E8 to U(1)×SU(2)×SU(3), the symmetry group of strong and electro-weak interactions that is the basis of the Standard Model of particle physics. E6 has 78 roots, of which six are simple and 72 are non-zero roots. The primary division:

240 = 72 + 168

displayed in the properties of sacred geometries (see under the heading "240 = 72 + 168" in The holistic pattern) expresses the fact that they embody E8, whilst equally holistic subsections of these geometries embody its exceptional subgroup E6. It appears in the types of 8-tuples representing the 240 non-zero roots of E8 shown in the diagram opposite.