Non-Euclidean Spaces
Euclidean Geometry is the basic geometry that you can graph and plot points in. Euclidean geometry follows five different rules proposed by Euclid. The most basic example of Euclidean geometry is a plane that can be visualized by a flat piece of paper.
It is possible to draw a straight line segment from one point to another
ex. On a plane, no matter where you place two points, you can always connect them with a straight line. If you took a curved surface (surface of a sphere), then no matter where you put two points, the line will always be curved.
You can indefinitely extend any line segment continuously in a straight line.
ex. You can do this on a plane however, on a graph with a hole in it, you could not continuously and indefinitely extend a straight line segment
You can draw a circle with any straight line segment using it as your radius and one of the endpoints being the radius
All right angles are congruent
Starting with a line and any point not on that line, there is only one line that can be drawn through that point that will be parallel to that original line
This can also be said that for any two lines, if you draw a third line across them, for any same side interior angles that are not 90 degrees, extending the original lines will result in a triangle.
Now that we have established what the rules for Euclidean geometry are, let’s now look at what would fail these. The most basic example would be the surface of a sphere. One unique property of the sphere is that you can form a triangle with three right angles. Because of the sphere’s unique surface, starting with a straight line and turning 90 degrees the same direction at each corner will result in a three-sided figure or a triangle.
The most important non-Euclidean geometry is that of Hyperbolic. When using Euclidean geometry, it is visually logical. Parallel lines look parallel and right angles look 90 degrees. Hyperbolic geometry breaks away from this visual constraint. It focuses more on true definitions of terms as they are not always visually logical. Hyperbolic geometry is also continually growing as mathematicians learn more and more.
When more closely defining Hyperbolic geometry, it is known that it rejects Euclid's 5th postulate. Instead, for every one point not on a line, there are at least two lines parallel to that original line. Just like the sphere, where the sum of all angles in a triangle is not equal to 180 degrees, in Hyperbolic geometry the sum of the angles of a triangle is less than 180 degrees. Below is an image that clearly illustrates most differences in different geometries:
This hyperbolic idea leads way to general relativity and Riemann surfaces. One way to visualize a type of hyperbolic geometry is by looking at the inside of a sphere. Shapes closer to the center of the sphere appear normal however the closer to the surface, the more distorted the shape gets. This same idea applies to hyperbolic geometry. Although Hyperbolic geometry is not the most tangible of geometries, it is fundamental in understanding and defining mathematical terms.