The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky’s geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky’s writings its role in Lobachevsky’s geometrical the...

The dualistic structure of statistical manifolds in information geometry yields eight types (possibly mixed type) geodesic triangles passing through three given points, the triangle vertices. interior angles can sum up to $$\pi $$ like Euclidean/Mahalanobis flat geometry, or exhibit otherwise angle excesses defects. In this work, we initiate study dually spaces, termed Bregman manifolds, where ...

It is shown that the generalized geometries may be obtained as a deformation of the proper Euclidean geometry. Algorithm of construction of any proposition S of the proper Euclidean geometry E may be described in terms of the Euclidean world function σE in the form S (σE). Replacing the Euclidean world function σE by the world function σ of the geometry G, one obtains the corresponding proposit...

Einstein’s general theory of relativity fundamentally changed our view about the physical world. Different from Newton’s theory, Einstein’s space and time are not flat but can be warped by matter. For a curved space such as Einstein’s space, Euclidean geometry is no longer suitable, and Riemannian geometry is usually used instead. In parallel with physics, due to an explosion of data from all f...

In this paper we demonstrate how the geometrically motivated algorithm to determine whether a twogenerator real Möbius group acting on the Poincaré plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division algorithm to real numbers that represent hyperbolic distances. In the case that t...