 Euclid lived in the Greek city of Alexandria in Egypt around 2,300 years ago. He spent his life studying and teaching geometry. He published his ideas in a book called Elements. To this day, it is the foundation for our understanding of geometry and mathematical processes in general. From the early to mid-1800s, new geometries were studied by mathematicians like Johann Karl Friedrich Gauss, one of the greatest mathematicians of all time, and his colleague, George Friedrich Bernard Riemann. We'll cover a bit of what they found. To understand the differences between Euclidean geometry and other possibilities, we start with points and connect them with lines. The shortest distance between two points is the line with the least curves. In Euclidean geometry, this is a straight line. In this case, the least curvature is no curvature at all. Another name for the shortest line between two points is the geodesic. If we draw geodesics that are each perpendicular to a third line, they will be parallel to each other. They will never cross, even if they are extended to infinity. This is a key characteristic of flat Euclidean space. What we are talking about here is intrinsic characteristics of the geometry. Things like the sum of the angles of a triangle are 180 degrees, and the circumference of a circle is 2 pi times its radius. We can bend this two-dimensional surface into a third dimension and give it the look of curved space, but this curvature is extrinsic by nature. The intrinsic curvature is still flat. Parallel lines remain parallel. The sum of the angles of a triangle is still 180 degrees, and the circumference of a circle is still 2 pi times its radius. But there are other possibilities. One possibility for a geometry supposes that the parallel geodesic lines are diverging, getting further apart. As if space were being stretched between the lines, and the further up the lines you go, the more the space is stretched. The best example of this is the surface of a saddle, or a potato chip. Here, the sum of the angles of a triangle is less than 180 degrees, and the circumference of a circle is more than 2 pi times its radius. This is hyperbolic geometry. It represents space with a negative curvature. Another possibility for a different geometry supposes that the parallel geodesic lines are converging, and will eventually meet. It's as if space were being compressed between the lines, and the further up the lines you go, the more the space is compressed. The best example of this is the surface of a sphere, like the Earth itself. Here, the sum of the angles of a triangle is greater than 180 degrees, and the circumference of a circle is less than 2 pi times its radius. This is spherical geometry. It represents space with a positive curvature. Here, the curvature is constant throughout the surface. The baseline is the equator. Parallel to the equator, we have the lines of latitude. Perpendicular to the equator, we have lines of longitude. They meet at the north pole. Remembering that three points define a plane, here's the best way to find the geodesic between two points on the surface of a sphere. First, draw the triangular plane defined by these two points, plus the center of the sphere. Then extend it to intersect the sphere's surface. The intersection creates a line around the sphere, called the great circle. The portion between the two points is the geodesic, the shortest line on the surface connecting the two points. A parallel of latitude line between the two points would be longer. This is why planes in the northern hemisphere travel north and then back south to get to a destination at the same latitude rather than travel due west or due east. They have chosen the shortest distance between the two points to save on both time and fuel. Some geodesic routes can save up to 1,000 kilometers. Of course, when it comes to a more generalized curved space, the curvature changes from place to place. If we were to put a car on a curved surface like this one and lock its steering wheel to go straight, it would naturally follow the space's geodesic line from its starting point. When it's done, it will have traveled the shortest distance between its starting point and its ending point.