Editing Non-Euclidean geometry (section)

==Kinematic geometries==
Hyperbolic geometry found an application in [[kinematics]] with the [[physical cosmology]] introduced by [[Hermann Minkowski]] in 1908. Minkowski introduced terms like [[worldline]] and [[proper time]] into [[mathematical physics]].  He realized that the [[submanifold]], of events one moment of proper time into the future, could be considered a [[hyperbolic space]] of three dimensions.<ref>Hermann Minkowski (1908–9). [[s:Space and Time|"Space and Time"]] (Wikisource).</ref><ref>Scott Walter (1999) [http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf Non-Euclidean Style of Special Relativity]</ref>
Already in the 1890s [[Alexander Macfarlane]] was charting this submanifold through his ''Algebra of Physics'' and [[hyperbolic quaternion]]s, though Macfarlane did not use cosmological language as Minkowski did in 1908. The relevant structure is now called the [[hyperboloid model]] of hyperbolic geometry.

The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the [[split-complex number]] ''z'' = e<sup>''a''j</sup> can represent a spacetime event one moment into the future of a [[frame of reference]] of [[rapidity]] ''a''. Furthermore, multiplication by ''z'' amounts to a [[Lorentz boost]] mapping the frame with rapidity zero to that with rapidity ''a''.

Kinematic study makes use of the [[dual number]]s <math>z = x + y \epsilon, \quad \epsilon^2 = 0,</math> to represent the classical description of motion in [[absolute time and space]]:
The equations <math>x^\prime = x + vt,\quad t^\prime = t</math> are equivalent to a [[shear mapping]] in linear algebra:<math>\begin{pmatrix}x' \\ t' \end{pmatrix} = \begin{pmatrix}1 & v \\ 0 & 1 \end{pmatrix}\begin{pmatrix}x \\ t \end{pmatrix}.</math>
:
With dual numbers the mapping is <math>t^\prime + x^\prime \epsilon = (1 + v \epsilon)(t + x \epsilon) = t + (x + vt)\epsilon.</math><ref>[[Isaak Yaglom]] (1979) A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity, Springer {{isbn|0-387-90332-1}}</ref>

Another view of [[special relativity]] as a non-Euclidean geometry was advanced by [[Edwin Bidwell Wilson|E. B. Wilson]] and [[Gilbert N. Lewis|Gilbert Lewis]] in ''Proceedings of the [[American Academy of Arts and Sciences]]'' in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into [[synthetic geometry]] of premises and deductions.<ref>[[Edwin B. Wilson]] & [[Gilbert N. Lewis]] (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the [[American Academy of Arts and Sciences]] 48:387–507</ref><ref>[https://web.archive.org/web/20091027012400/http://ca.geocities.com/cocklebio/synsptm.html Synthetic Spacetime], a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by [[WebCite]]</ref>