π
We define a vacuum point with a bivector (u, -u) in Ex(-E), ||u||=1 on each of its infinite orthogonal dimensions (this point is static because ui+(-ui)=0), with the anti-symmetry rule u+- u=0 and the orthogonality rule u.v=0 for all of its bivectors. Then a 1D direction called «1» (the 1 vector (u0, -u0) = (1, -1)) is chosen among them; a fundamental particle P is a vacuum point with a local anti-symetric permutation p (called superimposition or elementary orthogonal rotation of a 2D orthogonal base or the elementary orthogonal phase or, simply, the phase) switching two orthogonal directions u and v whose plane contains 1, p(u)=v, p(v)=-u such that u and v, v and -u superimpose (u+v and v-u) to let P to exist on its own, meaning that the introduced phase is external, as a local phase compensated by adjacent phases. Because of this superimposition, the 1 bivector becomes then a 1/2 bivector (u1, -u1), ||u1||=1/2, as the linear combination of the two 1/2 orthogonal vectors u/ 2 and v/2. We understand then that the 1 bivector is a dimensional singularity giving, in other terms, the direction of the local orthogonality as a local fusion of u/2 and v/2 along the axis of symmetry u1 between these vectors. The space where the study is about the only vector 1, meaning a simple vector space with no rules, no phase, is purely annoying and can be forgotten as a mathematical artefact, to keep the 1 length to the ½ vector. The scenario is then to consider an infinite space where local non zero phases will identify particles, separated by phases on their common dimensions. Instead of (u, -u) (v, -v), the switched vacuum point is now defined by (u, p(u)) (v, p(v))=(u, v) (v, -u) or (u, -v) (v, u): u+v is orthogonal to v-u, both non zero. The vacuum point is then seen with two halves entities with non zero speed. In other terms several parts of a vacuum particle gain initial partially opposite or orthogonal speeds because the geometry is locally partially torn. The analogy is to say that an axis is seen as a static point in its direction, and as a flux if oriented. Because of the switch, we consider that rules are also switched locally (u.(-u)0 and u+v0) while the local geometry incurs regular rules, ie a particle is a geometrical and dimensional singularity. A phase is introduced with (u’, v’) to get u2, ||u2||=1/2, where (u’, v’) is orthogonal to (u, v), u1 is the symmetric of (u’, v’). A phase is introduced with (u’’, v’’) to get u3, ||u3||=1/3, where (u’’, v’’) is orthogonal to (u, v) and (u’, v’), u1 is the symmetric of (u’’, v’’). With 3D, two orthogonalities are enough to define an inner 2D-cycle as a combination of permutation and an open cycle as a combination between a 2D-cycle and permutation with a third orthogonal direction, this one being any other orthogonal dimension to the 1 vector. Other dimensions (more than 3) does not give more freedoms to orthogonality and anti-symmetry because the result is a composition of orthogonality or anti-symmetry, being a combination of 1D (anti- symmetric switch between u and -u), 2D (idem and/or inner orthogonal switch), or 3D (idem and/or combination, and/or external orthogonal switch) permutations. A phase is introduced to get un, ||un||=1/n where n is the number of particles under the horizon of the particle u1. th Recursively, to build a n dimension un, another couple (u, v) orthogonal to ui, i