Euclidean Geometry was once considered a perfectly consistent and complete mathematical system.

The emergence of Non-Euclidean Geometries, like hyperbolic and elliptic geometry was an intellectual revolution. It turns out that Euclid’s Parallel Axiom is incomplete. It was an assumption about a line would far beyond our experience. What if we change the rule? What happens then?

In the 18th and 19th centuries, mathematicians attempted to correct newly perceived flaws in the parallel axiom.

Some attempted to prove that the parallel axiom was unnecessary and parallels could be derived from Euclid’s other axioms. This was incorrect. The other axioms do not lead to parallels, and without the parallel axiom they mathematicians would get wierd results that contradict many of Euclid’s theorems.

Some of the greatest mathematicians of the 19th century worked on the parallel problem, including Karl Gauss. Gauss rejected the parallel axiom and discovered hyperbolic geometry.

Euclid‘s 5 basic axioms:

1: Any two points can be joined by a straight line.
2: Any straight line segment can be extended indefinitely in a straight line.
3: Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4: All right angles are congruent.
5: Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Playfair tried to create a substitute parallel axiom:
“Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.”

This did not perform any better.

The fifth axiom was problematic. But so too was the second axiom. Both presumed that a line continued in an infinite direction. Euclidean Geometry was viewed as a complete and consistent mathematical system for two thousand years. The axioms were seen as self-evident and true. It turns out there is less than meets the eye. The definitions behind the axioms are assertions without evidence, and there is far greater incompleteness than first supposed.

For instance, what does the second axiom mean by a line extending indefinitely? Traditionally, it was defined as an infinite straight line. Elliptic Geometry offers an alternative definition. A line may be endless but finite. Imagine traveling along the equator on a sphere. You may circle the equator endlessly but the distance of the equator is finite. It is finite and unbounded at once. In Elliptical Geometry there are no parallel lines. The axiom is done away with entirely and the mathematicians can work with a new geometry. This creates Spherical Geometry and Prospective Geometry, amongst others.

Hyperbolic Geometry also questions the parallel axiom by asserting that there are an infinite number of parallel lines.
(image from wiki)

As science progressed in the 19th century, many realized that Euclidean Geometry did not truly describe the real world. They realized that Euclid’s Parallel definition was only one of many possible alternatives.

There are interesting new theorems in the new geometries that contradict the Euclidean geometry. Under the axioms of the new geometries we deduce that the sum of the angles of a triangle will never equal 180 degrees.

Gaussian Hyperbolic Geometry proved that the sum of the angles of a triangle is always less than 180 degrees. Odder still, as the triangle’s area decreases, the sum of the angles increases closer to 180 degrees.

Reimann’s Spherical Geometry proves the opposite. The sum of the angles of a spherical triangle is always greater than 180 degrees. As the area of the triangle decreases, the sum of the angles decreases towards 180 degrees.

So why is this useful?
Prospective Geometry (one form of Elliptic) is mostly used for art from what I know. To be honest, a lot of non-euclidean geometry produces fancy art but isn’t of much more use.

More pragmatically, Spherical Geometry is necessary for circumnavigation.

Circumnavigating over very short distances can be measured through Euclidean geometry because the spherical curves are not a major factor. So if you build a fence in you backyard, you can use Euclidean geometry to pre-measure the lengths and angles with a low margin of error. However, transoceanic travel over vast distances which require precise calculation of latitude and longitude which relies Non-Euclidean Spherical Geometry.

Hyperbolic Geometry is perhaps the most useful for physics.

In the past, scientists believed that two points in space could be connected with a straight line from point A to point B like a piece of taunt string.

It turns out this is incorrect. Einstein’s theory of relativity uses hyperbolic geometry to describe curved space. The force of gravity creates a curved gravity well in space-time. Any object creates its own gravity well, bending space-time as it moves. Spacetime is measured on a four-dimensional axis of x, y, z, and t so we can measure the changes of the space curve over time.