Another Constructive Axiomatization of Euclidean Planes

Abstract

Abstract. H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic = 2 in languages containing only operation symbols.Mathematics Subject Classification: 03F65, 51M05, 51F15, 03B30.Keywords: Constructive axiomatization of Euclidean planes, Constructive geometry. [6 -8] the following question: Which points in the Euclidean plane can one construct with ruler and compass, if one starts with a certain finite set of points, and the ruler is to be used for drawing the line joining two already constructed points and finding the intersection point of two already constructed lines, and the compass is to be used to draw uniquely determined points of intersection of circles and lines or circles and circles ? In other words, one does not consider the two intersection points (if they exist) of a line determined by two points A and B with a circle C, with neither A nor B lying on C, to be constructed. The reason is twofold: (i) one does not know whether C will actually intersect the line joining A with B, as this depends on whether the distance from the centre of C to the line joining A with B is less than or equal to the radius of C; (ii) even if one knew that they do intersect, in case there are two distinct intersection points, one would be unable to separate them by metric properties alone (these being the only properties that pure ruler and compass constructions, which do not involve the choice of one point from a set of two, can discern), without taking recourse to betweenness considerations, so one could not say that any of the two points was uniquely determined. For the same reasons one does not consider the points of intersection of two circles to be constructed, unless one of them is an already constructed point (in which case the second point is uniquely determined even if it coincides with the first one). H. Tietze investigated inHe showed that the set of points uniquely constructible by ruler and compass from a given set of points coincide with those constructible by ruler and set square, where the set square is used to drop a perpendicular from a point to a line and to raise a perpendicular from a point on a line to that line. 1)

Axiomatizing geometric constructions

“…Further constructive axiomatizations for Euclidean planes are possible, as shown in [86], in the bi-sorted languages…”

“…Non-elliptic metric planes have been constructively axiomatized in [92] in the language L(a 0 , a 1 , a 2 , F, π) with individual variables to be interpreted as points, with a 0 , a 1 , a 2 , F having the usual interpretations, and with π a ternary operation symbol, π(abc) being interpreted as the fourth reflection point whenever a, b, c are collinear points with a = b and b = c, an arbitrary point, otherwise. By fourth reflection point we mean the following: if we designate by σ x the mapping defined by σ x (y) = σ (xy), i.e.…”

Groups and Plane Geometry

“…That all axioms of B are satisfied with these definitions is now a matter of verification, for the most part accomplished in [7].…”

The Steiner–Lehmus theorem and “triangles with congruent medians are isosceles” hold in weak geometries

“…The theory of metric planes has been studied intensely in [5], where two axiom systems for it can be found. Two other axiom systems were put forward in [27] (for the non-elliptic case only) and [28].…”