Non-Archimedean geometry

The totality of geometrical propositions that can be deduced from the following groups of axioms: incidence, order, congruence, and parallelism, in Hilbert's system of axioms for Euclidean geometry, and that are unrelated to the axioms of continuity (Archimedes' axiom and the axiom of completeness). In a narrower sense, non-Archimedean geometry describes the geometrical properties of a straight line on which Archimedes' axiom is not true (the non-Archimedean line).

To investigate geometrical relationships in non-Archimedean geometry, one introduces a calculus of segments — a non-Archimedean number system, regarded as a special number system. One defines the concepts of a segment, and the quotient, the sum and the product of two segments. In particular, one introduces a Desarguesian number system — a non-Archimedean ordered field. With the aid of these number systems one constructs a theory of similarity of figures, a theory of areas, etc. The theory of areas of polygons underlying the theory of measurements of areas in a non-Archimedean plane is based on the concept of isometry of polygons with respect to completion, which is a more general concept than that of isometry with respect dissection.

In non-Archimedean geometry there exist triangles with equal heights and bases that are isometric with respect to completion but not to dissection. Isometric polygons with respect to completion have the same measure of area, and two polygons with the same measure of area are always isometric with respect to completion. Pythagoras' theorem for right-angled triangles is valid in non-Archimedean geometry.

The calculus of segments is used to introduce a system of affine (or projective) coordinates in a non-Archimedean space. For example, one selects in the plane two straight lines — coordinate axes — passing through a fixed point, and then lays off equal segments on each. In this affine coordinates system, the equation of a straight line is linear, i.e. of the form $ax+by+c=0$, where $x,y$ are numbers (segments) defining the coordinates of points on the line and $a,b,c$ are fixed numbers (segments).

The construction of numerical models for non-Archimedean geometry leads to what are known as Hilbert's transfinite (non-Archimedean) spaces. A number space of this kind on the real line is called a linear Veronese space.

The numerical realization of a non-Archimedean geometry, in which the commutative law of multiplication is not necessary, also plays an important role in the construction of Non-Desarguesian geometry. Such a geometry must be based on axioms of incidence, order and parallelism, without congruence.

The significance of non-Archimedean geometry lies in its role in investigating the independence and consistency of Hilbert's axioms for a Euclidean space. The realization of the axioms of incidence, order, congruence, and parallelism in a numerical model proves both that they are independent of the axiom of completeness and that non-Archimedean geometry itself is consistent. On the other hand, the role of the axioms of continuity in the construction of Euclidean geometry on the basis of Hilbert's axioms is also clarified. In particular, without the axioms of continuity one cannot prove the equivalence of Euclid's axiom of parallelism to the proposition that the sum of the interior angles of any triangle is equal to two right angles.

Geometrical constructions on the non-Archimedean plane are always carried out with the aid of a straight-edge with a marked-off standard of length (marked-off segment).

References

 [1] D. Hilbert, "Grundlagen der Geometrie" , Springer (1913)