However in the situation when the strict boundary accuracy is required and the global model of the Earth's surface is considered (long-range navigation, cartographic projections in small scales, astronomy) the Euclidean geometry becomes not sufficient for the geometric description and the analytic calculations coming from it.

Topics include Euclidean geometry, toy geometries and main definitions, discrete subgroups of the isometry group of the plane and tilings, the Poincare half-plane model, hyperbolic trigonometry and absolute constants, and the hierarchy of geometries.

To those who are used to Euclidean geometry such as taught in school, this relatively new branch of mathematics, bulking large in the science meetings, will seem strange.

It is also known as a type of non- Euclidean Geometry, being in many respects similar to Euclidean Geometry.

We all know that Euclidean geometry does not adequately explain the space in which we live.

18) Geometry, by contrast, is a priori and synthetic, requiring more than merely "conceptual thought," namely, a nonlogical capacity through which we intuit the self-evidence of the axioms of Euclidean geometry.

Before hyperbolic geometry was discovered, it was thought to be completely obvious that Euclidean geometry correctly described physical space, and attempts were even made, by Kant and others, to show that this was necessarily true.

Whereas Euclidean geometry consists of smooth, straight lines, fractal geometry consists of rough or fragmented shapes that can be split into parts, each of which is a reduced-size copy of the whole.