The proto-octomidial-axiom and nonstandard Lobachevsky geometry

In Euclid geometry the angle-sum of a triangle is 180 degrees. In Lobachevsky geometry the angle-sum is always less than 180 degrees

In order to make sure that a planar geometry is not empty one includes the axiom that there are two different proto-points say o and e . For the equilateral proto-triangle o e t one  constructs the top point t with the proper orientation.

In Lobachevsky geometry every equilateral triangle has an angle that depends on the length of its sides. and for every admissible angle there is a unique side length of the corresponding equilateral triangle. This means that one cannot choose just any proto-pair o and e . On the other hand one can construct right angles (90 degrees) and octomidial angles (45 degrees) . One chooses the proto-pair as a standard such that the corresponding equilateral proto-triangle has three angles with 45 degrees.

One can combine 8 equilateral proto-triangles to give an octagon, that's where the name of the following proto-octomidial  axiom is taken from. For an understanding the syntax of the axiom with the ternary isoscition function you have to study some sections of 'Geometries of O' . 

However, one is free to take another regular polygon to base Lobachevsky geometry on. Take e.g. the  regular heptagon. As one cannot construct the regular heptagon in the Klein-model of Lobachevsky geometry the heptamidial Lobachevsky geometry is different from octomidial Lobachevsky geometry, it is a nonstandard Lobachevsky geometry . This is more than just a convention as every nonstandard Lobachevsky geometry (and there are infinitely many) contains the standard Lobachevsky geometry. This follows from Liebmann's Theorem, that one can construct a triangle from its three angles. But this is may be leading a little bit too far for the moment ...