Rolling simplexes and their commensurability. (Axiom and criterion of incompressibility, momentum lemma)

Abstract

The total geometrical theory of field is recreated. DOI: 10.3103/S0027132211050093 “Cogito ergo sum.” Rene Descartes It is hard to establish now who first noticed that Newton’s first law can be interpreted as Kepler’s second law for any observer positioned out of the line along which a body moves freely and who tried to restrict himself with axiomatization of spatial properties of natural geodesics in the basic axiomatics of the classical and celestial mechanics in order to keep off the usage of Euclidean metrics. However, there is no doubt that each new generation continues constructions of new variants of the theory of gravitation based again and again on the mathematical apparatus of (Euclidean, Hilbert, etc.) metric spaces, paying no attention on the “golden rule of mechanics” and the “level arm rule”, and demonstrating the persistence deserving a better cause. In this paper we present other (and, in our opinion, weighty) arguments in favor of the moderation in all activities, in particular, it is worth replacing compasses by a set square in solution of construction problems at a certain stage of teaching plane geometry. Let the structure of an abstract projective plane be given on the set M , i.e. (see [1]), a subset L is chosen in the set 2 of all subsets of the set M and it is accepted to call the elements l of this subset straight lines satisfying the following properties (axioms): (P0) each line contains not less than three points; (P1) exactly one line l ∈ L passes trough any two points X,Y ∈ M ; (P2) any two lines l1, l2 ∈ L cross exactly at one point. For any straight line l ∈ L we assume Ml def = M \ l, Ll def = L \ {l} and, as always, call Ml with the system of lines Ll the affine map of the projective plane M , and call l the infinitely distant line. The lines l1, l2 ∈ Ll of the affine plane Ml are parallel if the point of their crossing lies on the infinitely distant line l. Proposition 1. Let the points A,B,C of the affine plane Ml do not lie on the same line. Then any three points A′, B′, C′ can be rolled in a finite number of steps (preserving the “area of ΔA′B′C′” in the intuitive sense)1 so that the result of this rolling of the points A′, B′ coincides with A and B, respectively, and the point C′ appears on the same line with the points B and C. Obviously, in the ordinary model of the affine plane studied in the school course of the planimetry, the following assertions are valid for the rolling process considered here. (RO) (Axiom of incompressibility). If the points A,B,C of the affine plane Ml do not lie on the same line and the point D lies on the same line with B and C, but not coincides with C, then the points A,B,D cannot be rolled to the points A,B,C, respectively. (R1) (Axiom of weak additivity). If the triples of points S,A,A′ and S,B,B′ lie on different lines in the affine plane Ml and the lines passing through A,B and A′, B′ are parallel, then the points S,A′, B can be rolled to the points S,A,B′, respectively. 1At the rolling step we “move” any point from the ordered triple parallel to the line passing through the remaining two points.