Axioms of motion (Redei version)

Question

I try to understand the axioms of motion in Redei's "Foundationd of Euclidean and Non-Euclidean Geometries, according to F. Klein" 1968

Redei gives as Axioms:

1. Any motion is a one to one mapping of space R onto itself such that every three points on a line will be transformed into (three) points on a line.

2. The identical mapping of space R is a motion.

3. The product of two motions is a motion.

4. The inverse mapping of a motion is a motion.

5. If we have two planes A, A' two lines g, g' and two points P, P' such that $ P \in g \subset A$, $ P'\in g' \subset A' $ then there exist a motion mapping A to A', g to g' and P to P'

6. There is a plane A, a line g, and a point P such that $ P \in g \subset A$ then there exist four motions mapping A, g and P onto themselves, respectively, and not more than two of these motions may have every point of g as a fixed point , while there is one of them (i.e the identity) for wich every point of A is fixed.

7. There exists three points A, B, P on line g such that P is between A and B and for every point $ C \ (C \not = P) $ between A and B there is a point D lying between C and P for which no motion with P as fixed point can be found that will map C onto a point lying between D and P.

8. There exists a line g such that $ g \cap R' \not = \emptyset $ having the following property: If A,B,C are three points of $ g \cap R'$ and B is between A and C , and M is a motion that $ A^M = B . (A^{M^2} = ) B^M = C $, mapping the set of points $y$ of points between onto the set of points between B and C: then every point of g is either one of the $ A^{M^i} $ or a point of the $ y^{M^i} $ $ ( i \in Z)$

As I understand it

Axioms 1 and 5 mean that there is a motion from every line to every line

Axioms 2 to 4 that motions form a group

But then

Axiom 6 : there are 8 motions mapping A, g and P onto themselves (we are working in 3 dimensional space so have 3 degrees of freedom) or are only the direct (orientable) transformations, motions according to this axiom?

Axiom 7: cannot follow it

Axiom 8: cannot follow it

Can anybody give some clarity in this?

This is the only set of axioms of motion I know, maybe there are simpler sets, maybe this one is more complicated because it is based on three dimensional geometry not on plane geometry.

The only other set of axioms of motion I know (Ewald "Geometry, an introduction" 1971) allready assumes the idea of isometry. (while isometry should follow from motion not the other way around)

Answer 1

Hint

At first sight, it seems to me that it a sort of "revised" version of Hilbert's axioms for geometry (original version: Grundlagen der Geometrie [1899], Engl.tr. The Foundations of Geometry ).

Hilbert axiomatization is organized with three primitive "objects":

point, line, plane;

and three primitive relations:

Betweenness: a ternary relation linking points, Lies on (Containment), Congruence.

Axioms are grouped into:

I.Incidence II. Order III. Congruence IV. Parallelism V. Continuity.

If we want to axiomatize absolute geometry we have to discard the Parallels axiom: in this way we get the "common" part between Euclidean and non-Euclidean geometries.

It seems to me that in László Rédei's book:

• Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein (1968 - German ed.1965)

among other "minor" changes, the basic difference with respect to Hilbert's approach is to replace the primitive concept of congruence (and its axioms set) with that of motion.

This approach was introduce in 1900 (indipendently from Hilbert) by Mario Pieri in his: Della geometria elementare come sistema ipotetico deduttivo: Monografia del punto e del moto (On elementary geometry as a hypothetical deductive system: Monograph on point and motion).

See:

• Elena Anne Marchisotto & James Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (2007).

Answer 2

Pieri only introduces postulates as they become necessary in this development. So if you read the axiomatization itself (which is beautiful), you could see that Pieri needs this postulate 9 to be able to make an assertion about planes similar to this theorem he had proved for lines: If a and b are distinct points, if it happens that c and d are distinct points of the line ab, then the lines ab and cd will coincide

Regarding Pieri's postulation of the relation of being a subset: Pieri postulates only one way containment because this is just enough (coincidence being a natural consequence). Characteristic of all his axiomatizations, Pieri wants to only postulate the minimum of what is needed. He helps the reader see this. In a comment after the postulate, Pieri says: Consequently, the plane abc will also be contained in the plane abd, for which reason the planes abc and abd will coincide.