Comparing Geometries

In mathematics, there are many broad and partially undiscovered areas of learning that are parts of many school curriculums. One of those topics is geometry. In schools, mainly Euclidean is the only geometry taught, but there are two other types as well. Geometry can be split into Euclidean, Spherical and Hyperbolic. Euclidean geometry was the basis of all geometry as it is known today. It was founded by Euclid and followed his five postulates, a straight line segment can be drawn joining any two points, given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center, all right angles are congruent, and if two lines are drawn which 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. The most famous of those five postulates was the last postulate. This made many mathematicians think about whether they could find a counterexample for it, paving the way for research into spherical and hyperbolic geometries. These three geometries contrast the similarities and differences of shapes, and are the basis for how any of them work on any plane.

            The first shape that has contrasting features in each of the geometries is a triangle. Triangles are a type of polygon. Over all three geometries there are a few points that are the same or similar. All triangles have three sides and three angles, all three sides are the shortest distance between the two points and they do not intersect except at the corners to make the angles, and finally all three types can have two congruent triangles. In Euclidean geometry, the angle sum is equal to 180°. This differs from spherical, in which the angle sum is greater than 180° and different from hyperbolic, which is triangles with an angle sum less than 180°. Also, in spherical geometry there can be up to three right or obtuse angles, but in Euclidean there is a maximum of one obtuse or right angle. Finally, in hyperbolic geometry, as the angle measure gets smaller, the triangle gets larger. This is different from spherical geometry, where the triangle gets larger as the angles get larger. Similarly in Euclidean geometry the area is not affected by the angle measures. Also, polygons have many similar and differing characteristics across the three geometries. In both hyperbolic and Euclidean, polygons have at least three sides, but in spherical, they only need at least two sides. All three geometries believe polygons have any angle measure. Polygons and triangles are two of the many shapes that reflect the differences between Euclidean, spherical and hyperbolic geometry.

            In addition to triangles and polygons, circles have many characteristics that are shared and many that are not shared. Some of the shared characteristics are all circles consist of one curved line that stays an equal distance from a center point and they all can be made. In spherical geometry, the largest circle is the great circle, then as the radius becomes larger than the great circle, the area of the triangle decreases. This does not apply to Euclidean or hyperbolic because they have an infinite amount of space for the circle’s radius and area to be as large as it wants. Another difference is on a sphere, there is no difference between the finite area and the infinite area because both are circles. Finally, in all three geometries, it is possible to have more than one congruent circle on a single plane. Circles are shapes that follow a specific set of requirements and do not have any straight lines. They are also one of the few shapes that can be formed in any geometry as long as curved lines are permitted.

            As stated prior, the fifth postulate of Euclid is the reason mathematicians searched for and found the other two types of geometry. The fifth postulate states that if there is a line and a point, there is only one line going through the point that is parallel to the line. In spherical geometry, there are no parallel lines because all lines are great circles which means they go through the center, meaning they all cross. In hyperbolic geometry, there are an infinite number of parallel lines crossing through a point that is parallel to another line. This is because the raised planes allow for the lines to curve away from the original line. This means that the lines are not tradition parallel lines because they do not remain the same distance away from each other. Although it seemed impossible, mathematicians found a counter example that proved Euclid’s fifth postulate incorrect.

            In Euclidean, spherical and hyperbolic geometries, there are many characteristics of shapes and lines that show their differences and similarities. Across the world, people try to disprove statements and postulates as incorrect or correct. Most of the time this results in a new type of thinking that spreads rapidly across the world. In this case, the new thinking was the disproving of Euclid’s fifth postulate that spread the belief that other geometries were in the world and they held the answers about 3D planes that Euclidean could not answer.