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.