# Geometries of O

**Summary**

Geometries of O adhere to Ockham's principle of simplest possible ontology: the only individuals are points, there are no straight lines, circles, angles etc. , just as it was was laid down by **Tarski **in the 1920s, when he put forward a set of axioms that only contain two relations, quaternary congruence and ternary betweenness. However, relations are not as intuitive as functions when constructions are concerned. Therefore the **planar geometries of O** contain only functions and no relations to start with. Essentially three quaternary functions occur: **appension** for line-joining of two pairs of points, **linisection** representing intersection of straight lines and **circulation** corresponding to intersection of circles. Functions are strictly defined by composition of given ones only.

Both, Euclidean and Lobachevskyan planar geometries are developed using a precise notation for **object-language** and **metalanguage**, that allows for a very broad area of mathematical systems up to theory of types. Some astonishing results are obtained, among them:

(A) Based on a special triangle construction Euclidean planar geometry can start with a **less powerful ontological basis** than Lobachevskyan geometry.

(B) Usual Lobachevskyan planar geometry is not complete, there are **nonstandard** planar Lobachevskyan geometries. One needs a further axiom, the 'smallest' system is produced by the **proto-octomidial-axiom**.

(C) Real numbers can be abandoned in connection with planar geometry. A very promising **conjecture** is put forward, stating that the Euclidean Klein-model of Lobachevskyan planar geometry does not contain all points of the constructive Euclidean unit-circle.

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93 pages, version 1.1

first published for priority 05.12.2018 vixra 1812,0085