the principle of relativity: from ungar’s gyrolanguage for physics to weaving computation in mathematics

‎‎this paper extends the scope of algebraic computation based on a non standard times to ‎the more basic case of a non standard + ‎, ‎where standard means associative ‎and commutative‎. ‎two physically meaningful examples of a non standard + are ‎provided by the observation of motion in special relativity‎, ‎from either ‎outside (3d) or inside (2d or more)‎, ‎we revisit the ``gyro''-theory of ungar to present ‎the multifaceted information processing which is created by a metric cloth w ‎, ‎a relating computational construct framed in a normed vector space v ‎, ‎and based ‎on a non standard addition denoted pluscirc whose commutativity and associativity ‎are ruled (woven) by a relator‎, ‎that is a map which assigns to each pair of admissible vectors ‎in v an automorphism in aut w ‎. ‎special attention is given to the case where the relator is ‎directional‎.