The reconstruction of the third dimension

Seeing, the mind's processing of optical data, in essence is a mathematical construction which associates each set of identified two-dimensional objects on the retina with a three-dimensional (“real”) object. We call this construction the mental perspectival construction, or the reconstruction of the 3rd dimension. Mathematically, it is represented by the mapping

K_2,3: F₂ —› F₃,

where F₂ denotes the set of the 2-dimensional figures, and F₃ the set of the three-dimensional objects, our “real world.” We can divide the construction rules into two categories:

I. Naturally given construction rules. On the one hand there are rules for which the human culture is irrelevant:

1. (linear perspektive / ray theorem) Objects of the same size gets the smaller, the farer they are away. A magnification of the distance between object and observer by the factor a causes a diminutio of the object by the factor 1/a.
2. (Distance determination for connected surfaces) Two objects or parts of objects that are connected organically both are equally distant from the observer; they are interpreted as a connected object.
3. (Distance determinaion by blurness) Blurred figures, lines, and contours are interpreted to be farer away than ones of equal apparent size being unblurred.
4. (Orientation by brightness) Light sources are located at points which point against the directions of shadows and into which bright surface parts are pointing.

Besides there are reconstruction rules being determined by the conditions of sociocultural life. Here the right angle dominates. Our world is pressed into a rectangular coordinate system: It is a world of straight lines, plane surfaces and right angles.

1. (Surfaces) Eine surface is plane and continues as long as there are no hints interrupting the continuity.
2. (The three principal directions) In points where three identified planes meet, the three principal directions are determined.