Ketland, Jeffrey
(2023)
Length Abstraction in Euclidean Geometry.
[Preprint]
Abstract
I define abstract lengths in Euclidean geometry, by introducing an abstraction axiom: $\lambda(a,b) = \lambda(c,d) \iff ab \equiv cd$. By geometric constructions and explicit definitions, one may define the \emph{Length structure}: $\Len = (\Len, \oplus, \preceq, \bigcdot)$, ``instantiated by Euclidean geometry'', so to speak. I define the notion of a ``(continuous) positive extensive quantity'' and prove that $\Len$ is such a (continuous) positive extensive quantity. The main results given provide the general characterization of $\Len$ and its symmetry group (the multiplicative group of the positive reals); along with the relevant mathematical relationships between (abstract) lengths and \emph{coordinate} lengths (relative to a coordinate system); and also between lengths, measurement scales and units for length.
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