Around 300 BC, the Greek Euclid wrote "The Elements", which stated five postulates upon which he based his theorems. These postulates form the basis of what is known as Euclidean geometry, and is the foundation of the geometry most of us have studied.
Euclid’s postulates are:
1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
It has been obvious to many that Euclid’s fifth postulate differs from the first four, and many have sought to develop proofs of it. This has given rise to a study of shapes and constructions that do not map directly to any Euclidean system, or non-Euclidean geometry. Hyperbolic, or outward curving, and Elliptic, or inward curving, geometry, are two forms of non-Euclidean geometry. The non-Euclidean geometries modify the principle of Euclid’s fifth postulate.
Euclidean geometry, while the basic gospel of geometry until well into the 19th century, unfortunately does not adequately describe the action of shapes and constructions on certain surfaces, such as a ball, or inside a curved surface such as a cup, or even three-dimensional space. Debates about the applicability of Euclid’s theorems, debates which led to the development of non-Euclidean geometries, began almost as soon as The Elements was published. Theoroms developed by Ibn al-Haytham, Khayyam and al-Tusi, formed the basis of the non-Euclidean geometries.
The principal difference between Euclidean and non-Euclidean geometries is the nature of parallel lines. In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane and is parallel to the first line, never interesecting it. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to the original line. In spherical geometry there are no such lines. In spherical geometry, points are defined in the usual way, but lines are defined in a way that the shortest distance between two points lies along them. In a word, lines in spherical geometry are in fact 'great' circles, similar to the equator which divides the sphere of the earth into two equal hemispheres.
The development of non-Euclidean geometries, which increased in the 1800s, was critical to physics in the 20th century. Albert Einstein’s theory of relativity, for instance, which describes space as elliptically curved in areas where energy is present, owes its proof to non-Euclidean geometry, rather than the flat space of Euclidean geometry.
In fiction, especially the science fiction and horror genres, non-Euclidean geometry often plays a central role, such as in stories involving parallel universes or time travel.
While Euclid’s geometry is the study of flat space, it does not help to describe the action of objects within three dimensional or greater space, or in or upon a sphere. It has only been through the development of the non-Euclidean geometries that mankind has been able to understand the action of objects in other than two dimensions.