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We Define a Vacuum Point with a Bivector 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<n is needed, the continue transition from ui in n-1 dimensions to ui in n dimension being modeled by a continue component xn from 0 to 1 along un with ||un||=1/n such that (ui .(n-1)/(n-1+xn), xn.un) is the base transition from the base (ui<n) to the new nD base (ui≤n) where ui is then normalized by (n-1)n.ui = 1/n. P is then a dimensional construction with n orthogonal vectors to the 1 vector, called the multidimensional vector 1°=1+ui. Depending on the distance between two particles, meaning the phase between their common vectors, their relative phases characterize the scalar along their respective dimensions. Here we have considered that the vector 1 is common for each particle, however it is possible to give an initial orientation to each 1 vector and it becomes necessary to introduce actually nn phases. However, we have seen that only a combination of 1D to 3D permutations are enough to describe a distant interaction, ie we need only six 1° vectors with 6n phases, orthogonal, to define the phase between two 3D references: when these one are orthogonal, such in the case of the permutation i, j, k → j, k, i, where i, j, k are 1° vectors, the total length of the vector is 6.||1°|| where ||1°|| = (ui²) = π/6. A 1 vector is then accompanied by a π invariant. We interpret this result as the real measurement of the travel of such particle: this particle must be the photon, traveling a distance 1 while drawing a circle with diameter 1.
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