In 1482, Euclid’s Elements was published, presenting the first clear and logical framework for a geometrical system. This Euclidean Geometry, which was known simply as ‘geometry’, is an axiomatic system, in which all theorems are derived from a smaller number of axioms. Euclid gives five postulates on which this “flat” or “plane” geometry is based:
1. To draw a straight line from point to point.
2. To produce a finite straight line continuously in a line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. Through any given point can be drawn exactly one straight line parallel to the given line.
The fifth postulate of Euclidean geometry became known as the Parallel Postulate. The Parallel Postulate maintains that given a straight line, and a point not on the line, there is exactly one line through the given point parallel to the given line. In Euclidean geometry, lines that are at a constant distance from each other, even when extended to an infinite length, are parallel. Non-Euclidean geometry arises when the Parallel Postulate does not hold. The simplest example is the surface of a globe. The given straight line would circumscribe the globe, any line drawn through any point not on the line would intersect the given line as it too will circumscribe the globe. Non-Euclidean geometry is concerned with the geometry of curved surfaces, in contrast with the flat surfaces in Euclidean geometry. While Euclidean geometry plays a large role here on a relatively flat Earth, non-Euclidean geometry plays a crucial role in the Theory of Relativity and the geometry of space-time.
There are two main forms of non-Euclidean geometry, hyperbolic geometry and elliptical geometry. Hyperbolic geometry is concerned with saddle shaped surfaces, which have a negative curvature. In regards to the Parallel Postulate, there are at least two, if not an infinite amount of distinct lines parallel to the given line which can be drawn through a given point not on the given line. In elliptical geometry, which is concerned with the surface of a sphere (as in the example of the globe), no line is ever parallel to any other line. Contrary to hyperbolic geometry, elliptical surfaces have a positive curvature.
The realization that Euclidean geometry was not the only geometry in our world cemented the foundation for physical science in the 20th century. It may have taken hundreds of years, but finally, proof was in existence that we live not in a flat universe, but that we live in a universe that curves and bends in the presence of energy.
