How to Understand Non-Euclidean Geometry

In the third century B.C., the Greek mathematician Euclid explored the basic geometry that's still taught in schools today. It explains the relationships between (and rules of construction of) triangles, squares, circles and other geometric shapes. For almost 2,000 years, Euclid's treatise was the last word on the subject-until, that is, modern mathematicians began developing new, non-Euclidean types of geometry. Here's how to understand non-Euclidean geometry.

Instructions

1. Euclidean geometry is based on Euclid's 5 self-evident rules that he called postulates. In order, these postulates are:
   1) only one straight line can be drawn through two given points;
   2) any line segment can be extended toward infinity in any direction;
   3) given any length and any point, a circle can be drawn with the point as its center and the length as its radius;
   4) all right angles are identical; and
   5) given a line and a point, only one line can be drawn through the point that is parallel to the first line.

2. Mathematicians have long been suspicious of the fifth postulate. Ever since Euclid's time, mathematicians have wondered if the fifth postulate (also known as the "parallel postulate") was truly a self-evident proposition, or if it could somehow be derived as an implication of the other four postulates. In the nineteenth century, this suspicion came to a head, and some mathematicians explored what the results would be if the parallel postulate were dispensed with.

3. "Hyperbolic geometry" was the first non-Euclidean geometry to be developed. Although it's difficult to visualize in spacial terms, hyperbolic geometry operates according to a simple premise: There can be an infinite number of parallel lines through a point adjacent to a single line, with the direct consequence being that the three angles of a triangle must add up to less than 180 degrees (compared to exactly 180 degrees in Euclidean geometry).

4. "Elliptic geometry" can be considered the inverse of hyperbolic geometry. Much easier to visualize, elliptic geometry can be interpreted as pasting classical Euclidean shapes onto the surface of a sphere. In elliptic geometry, it's not possible to draw even a single parallel line through a point adjacent to a line, with the consequence being that the angles of a triangle add up to more than 180 degrees.

5. Non-Euclidean geometries aren't just theoretical. Amazingly, both hyperbolic and elliptic geometry apply to the four-dimensional spacetime of Einstein's relativity theory: elliptic geometry applies because matter "curves" the space surrounding it, and hyperbolic geometry applies because it describes the expansion of the universe. In fact, if mathematicians hadn't developed non-Euclidean geometries decades before, it's unlikely that Einstein could have invented relativity in the first place!