Squeeze mapping
Some insight into logarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area.
Both circular and hyperbolic angle generate invariant measures but with respect to different transformation groups.
[2] In 1688, long before abstract group theory, the squeeze mapping was described by Euclid Speidell in the terms of the day: "From a Square and an infinite company of Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same properties or affections of any Hyperbola inscribed within a Right Angled Cone.
The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups SO ⊂ SL – in this case SO(1,1) ⊂ SL(2) – of the subgroup of hyperbolic rotations in the special linear group of transforms preserving area and orientation (a volume form).
This insight follows from a study of split-complex number multiplications and the diagonal basis which corresponds to the pair of light lines.
Formally, a squeeze preserves the hyperbolic metric expressed in the form xy; in a different coordinate system.
This application in the theory of relativity was noted in 1912 by Wilson and Lewis,[4] by Werner Greub,[5] and by Louis Kauffman.
[6] Furthermore, the squeeze mapping form of Lorentz transformations was used by Gustav Herglotz (1909/10)[7] while discussing Born rigidity, and was popularized by Wolfgang Rindler in his textbook on relativity, who used it in his demonstration of their characteristic property.
[8] The term squeeze transformation was used in this context in an article connecting the Lorentz group with Jones calculus in optics.
According to Stocker and Hosoi, The area-preserving property of squeeze mapping has an application in setting the foundation of the transcendental functions natural logarithm and its inverse the exponential function: Definition: Sector(a,b) is the hyperbolic sector obtained with central rays to (a, 1/a) and (b, 1/b).
The answer is the transcendental number x = e. A squeeze with r = e moves the unit angle to one between (e, 1/e) and (ee, 1/ee) which subtends a sector also of area one.
Such transformations of pseudospherical surfaces were discussed in detail in the lectures on differential geometry by Gaston Darboux (1894),[16] Luigi Bianchi (1894),[17] or Luther Pfahler Eisenhart (1909).
[18] It is known that the Lie transforms (or squeeze mappings) correspond to Lorentz boosts in terms of light-cone coordinates, as pointed out by Terng and Uhlenbeck (2000):[13] This can be represented as follows: where k corresponds to the Doppler factor in Bondi k-calculus, η is the rapidity.
It also squeezes blue and green rectangles .
