About Weaving Computation in 2 and 3 Real Dimensions
This work revisits the two basic examples of nonstandard + (denoted ◦+) given in the CERFACS report TR/PA/11/27 . These unconventional “additions” are disymmetric in the addends and relative to a reference value λ > 0; they are (i) Einstein ◦+ in R3 and (ii) Poincar´e ◦+ in C. Each of them participates in a specific synthesis between + and * which is represented additively in (i) Special Relativity in R3 and (ii) conformality in the unit disc of C. These synthetic operations are naturally connected to (i) derivations in the imaginary part H of the quaternions H and to (ii) internalisations in C which are presented in (Chatelin 2012 ).
We show how the departure from commutativity for Einstein’s addition reveals a natural logic derived in 3D from nonlinear computation. The logic can be ambiguous if Einstein’s addition on real vectors is used with real or complex conjugate scalars. Such is the case when, for a nonzero vector x ∈ R3, x/ λ ∈]0, 2], λ > 0. It also reveals some qualitative aspects of the constructive potential of computation in H. This potential is the common computational source for the two different perspectives on Special Relativity which had been designed independently in 1905 by Einstein and Poincare.
Keywords: Complex numbers, quaternions, derivations, internalisations, weaving computation, organ, cloth, conjugator, commutator, relativity, conformality, natural logic for 3D-evolution.