  Let a physical event constitute a simple loop in spacetime. This in turn
calls for a generalized loop line element (= distance$^2$ between two
neighboring loops) capable of restoring, at the shrinking loop limit, the
special relativistic line element (= distance$^2$ between the two neighboring
centers-of-mass, respectively). Sticking at first stage to a flat
Euclidean/Minkowski background, one is led to such a preliminary loop line
element, where the role of coordinates is played by the oriented cross-sections
projected by the loop event. Such cross-sections are generically center-of-mass
independent, unless (owing to a topological term) the loop events are
intrinsically wrapped around a Kaluza-Klein like compact fifth dimension.
Serendipitously, it is the Kaluza-Klein ingredient which, on top of its
traditional assignments, is shown to govern the extension of Pythagoras theorem
to loop space. Associated with $M_4 \otimes S_1$ is then a 10-dim loop
spacetime metric, whose 4-dim center-of-mass core term is supplemented by a
6-dim Maxwell-style fine structure. The imperative inclusion of a positive (say
Nambu-Goto) string tension within the framework of Loop Special Relativity is
fingerprinted by a low periodicity breathing mode. Nash global isometric
embedding is conjectured to play a major role in the construction of Loop
General Relativity.
