Geometries for Possible Kinematics

The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are revealed. Almost all geometries fall into pairs. There exists $t \leftrightarrow 1/(\nu^2t)$ correspondence in each pair. In the viewpoint of differential geometry, there are only 9 geometries, which have right signature and geometrical spatial isotropy. They are 3 relativistic geometries, 3 absolute-time geometries, and 3 absolute-space geometries.Comment: 40 pages, 7 figure

Polarization-Induced False Colours

If the photoreceptors of a colour vision system are polarization sensitive, the system detects polarization-induced false colours. It has been hypothesized that egg-laying Papilio butterflies could use these polarizational colours as a cue to detect leaf orientation and to discriminate between shiny and matte leaves. In this chapter, we show that a shiny green surface with any orientation can possess almost any polarizational false colour under any illumination condition (for different solar elevations and directions of view with respect to the solar azimuth as well as for sunlit and shady circumstances under clear skies). Consequently, polarizational colours cannot unambiguously code surface orientation. Polarization sensitivity is even disadvantageous for the detection of surface orientation by means of colours. On the other hand, the colour changes due to retinal rotation can be significantly larger for shiny surfaces than for matte ones. Thus, polarizational colours could help polarization-dependent colour vision systems to discriminate between shiny and matte surfaces. Earlier it has been believed that a uniformly polarization-sensitive retina (UPSR)—in which receptors of all spectral types have the same polarization sensitivity ratio and microvilli direction—cannot detect polarization-induced false colours. Here we show that, contrary to this belief, a colour vision based on a UPSR is subject to polarization-related artefacts, because both the degree and the angle of polarization of light reflected from natural surfaces depend on wavelength. These findings are of general importance for polarization-dependent colour vision systems