UNIVERSAL REALITY
UNIVERSAL REALITY
In the previous section we have uncovered a real and an imaginary section of space-time that, while derived from equation [4], are both orthonormal to the vector space therein, hence, neither section is apparent in Lorentz space-time in the observer’s line-of-sight (their individual components will be evident in roles that have been explained above and further below). Next we will take an ostensible detour in which the nature of binocular observation will be reviewed; insight into the stereoscopic information it contains will lead to another section of the extended 3-space that will provide us a second set of real and imaginary planes.
Binocular vision is one of a number of observation modes collectively known as stereometry that provide a sense of depth (hearing is another, but lacking the equivalent of a ‘field of vision’ it relies on additional queues to give a sense of three dimensional aural ambience, not just depth). Our eyes acquire two plane ‘landscape’ views composed of unobstructed points at any distance in two overlapping sections of our surroundings by collecting all such lines that enter our field of vision through an optic lens (e.g. each of our eyes will collect these points in an approximately conical section, other types of lenses may cover a different section, such as cylindrical by a parabolic lens). The direction of the line-of-sight is determined by the line that passes through the lens without refraction and is the subject of the discussion that follows. Think of the line-of-sight as the axis of rotation (lying) in a second plane that is perpendicular to the plane view on which it is projected at any angle of polarization with respect to an arbitrarily chosen reference direction (such as the horizon). The cone or cylinder mentioned above is a section of the 3-space that will result from rotation of the second plane. Our brain reconstructs (or e.g. a computer triangulates) what appears to be a third dimension from these two normal plane (2-space) views. Both planes provide a view that is perpendicular to their line-of-sight and displaced with respect to each other: parallax The displacement in point of origin of the two planes- e.g. the distance between our eyes- is the base-line between observation stations. Although binocular vision provides us with two lines-of-sight, we cannot see the third dimension through either lens directly. All we can perceive is a (nearly) contiguous collection of infinitesimal points of the imagined third spatial dimension as if seen on end: the line-of-sight being the point of origin around which the normal plane we see with each eye can rotate axially. However, stereoscopy is confined to only two plane views (complex if additionally we consider the temporal component), and their parallax angle, thus missing two of the four planes (and their angular relationships) that observation of the extended 3-space requires; in terms of any sensory measurement: our macroscopic experience while stereometric is not three-dimensional.
How does the above relate to our aim of defining two real and two imaginary planes and their angular relationships? Binocular observation involves the measurements (in two frames) of spatial vectors s and s' that include a parallax angle, their relation can be expressed with the law of cosines applied to the parallax angle as s + s'. However, the orientation of the aforementioned vector s' is the same as s in binocular vision, e.g. away from the observer, and thus as measured will oppose that of s' in equation [4] where as previously noted s and s' are of opposite direction. Accordingly, in terms of the vectors in [4] and subsequent derivative equations, the binocular relation would be expressed as s + (-s') which is the very relation (s' - s) we encountered in equation [3].