In two papers titled On the so-called non-Euclidean geometry, I and II ([32] and [34]), Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) within the realm of projective geometry. Klein’s work was inspired by ideas of Cayley who derived the distance between two points and the angle between two planes in...

A non-contradictible axiomatic theory is constructed under the local reversibility of the metric triangle inequality. The obtained notion includes the metric spaces as particular cases and the generated metric topology is T$_{1}$-separated and generally, non-Hausdorff.

Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the spacetime Pythagoras theorem. Introduction Spacetime dia...

Second proof. Let O be the circumcenter of ABC and let ω be the circle centered at D with radius DB. Let lines AB and AC meet ω at P and Q, respectively. Since ∠PBQ = ∠BQC+∠BAC = 1 2 (∠BDC + ∠DOC) = 90, we see that PQ is a diameter of ω and hence passes through D. Since ∠ABC = ∠AQP and ∠ACB = ∠APQ, we see that triangles ABC and AQP are similar. If M is the midpoint of BC, noting that D is the m...

Non-Euclidean method of the generalized geometry construction is considered. According to this approach any generalized geometry is obtained as a result of deformation of the proper Euclidean geometry. The method may be applied for construction of space-time geometries. Uniform isotropic space-time geometry other, than that of Minkowski, is considered as an example. The problem of the geometric...

For more than two millennia, ever since Euclid’s geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time, a restriction which above all affects Physics. Here it is argued that, ever since the invention of Calculus by Newton, we may actually have empirical evidence that t...

Introduction 3 1. Affine geometry 4 1.1. Affine spaces 5 1.1.1. Euclidean geometry and its isometries 5 1.1.2. Affine spaces 7 1.1.3. Affine transformations 8 1.1.4. Tangent spaces 9 1.1.5. Acceleration and geodesics 10 1.1.6. Connections 11 1.2. The hierarchy of structures 11 1.3. Affine vector fields 12 1.4. Affine subspaces 13 1.5. Volume in affine geometry 14 1.6. Centers of gravity 14 1.7....