The World of Non-Euclidean Geometry

What's so special about non-euclidean geometry?

Being relatively inexperienced in the many different branch fields of mathematics, I was curious as to how this non-euclidean geometry was different from regular old Euclidean geometry. In order to understand non-euclidean geometry, we must first look at Euclid.

Euclidean Geometry

Euclid is reported to have lived sometime during the 4th and 3rd centuries BCE. He is known as, "the father of geometry." He is also credited with writing the Elements, in which he includes several assumptions. These encompassed his five postulates:

1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in a straight line."
3. "To describe a circle with any center and distance [radius]."
4. "That all right angles are equal to one another."

a. When the lines intersect, and the adjacent angles are equal, each angle

is said to be a right angle

The fifth postulate, however, is what gave rise to non-euclidean geometry:

The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

For centuries, mathematicians were puzzled by this postulate. It was complicated but they believed it could be proven from the other four. To attempt to prove it, they subjected it to a proof by contradiction. Their work was unsuccessful in proving the fifth postulate from the the other four, but their efforts were not in vain.

Non-Euclidean Geometry

Kinds of non-euclidean geometry arose from the attempted proving of the fifth postulate, but were not recognized as legitimate until the 19th century. The two main models of non-euclidean geometry that I researched are Elliptic geometry and Hyperbolic geometry.

Elliptic Geometry

From Wikipedia:

"Given a line L and a point p outside L, there exists no line parallel to L passing through p, as all lines in elliptic geometry intersect."

The easiest way to picture such a geometry is to look at a sphere:

Lines in elliptic geometry can be imagined as great circles because all lines in elliptic geometry intersect.

Another way elliptic geometry differs from euclidean geometry is the way figures can be scaled. In euclidean geometry, figures can be scaled up or down and not change shape, i.e. the shapes are similar. In elliptic geometry, however, this cannot be done. Wikipedia explains:

"For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified). A line segment therefore cannot be scaled up indefinitely."

The pythagorean theorem also fails in elliptic geometry. In elliptic geometry, the sum of the interior angles of a triangle is greater than 180 degrees. From Wikipedia:

Hyperbolic Geometry

From Wikipedia:

"For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. This implies that there are through P an infinite number of coplanar lines that do not intersect R."

These two images from Wikipedia can help with the visualization of the above and of how space works in hyperbolic geometry:

The following image is what a triangle would look like if placed in a saddle shape plane. It also includes two diverging parallel lines:

All images are from Wikipedia