## Euclidean, Spherical, Eliptic, and Hyperbolic geometry

Though in schools most students learn Euclidean Geometry, there are actually many different types. These different types were developed by other mathematicians who developed theories and research that may have contradicted the work of other

mathematicians. This resulted in the formation of four different types of geometry. The four types are Euclidean, Spherical, Eliptic(aslo known as Riemann’s geometry), and hyperbolic.(Also known as lobachevsky’s geometry)

Euclidean Geometry which is sometimes called “flat” or “parabolic” geometry is named after the greek mathematician Euclid of Alexandria. Euclid lived in alexandria, Egypt and is credited for writing Elements which is considered to be the most successful textbook in the history of mathematics. Also, he is known for his use of logical reasoning to prove mathematical theorems which is still the backbone to mathematics. One of his most credited accomplishments was his reasearch of Euclidean Geometry which is based almost entirely on his studies.

Euclidean Geometry is based on rules called postulates. It is different from other geometries because of the parralel postulate which states that through a point not on a given straight line, one and only one line can be drawn that never meets the given line. This is what seperates Euclidean geometry from other types such as Reimann geometry where no parralel lines exist. In Euclidean geometry each point is mapped on a rectangular coordinate system in Euclidean space with a unique set of real numbers. It is used to describe points in space or on a plane to express geometric relations.

Spherical geometry contradicts Euclidean geometry in two ways. Spherical geometry states that there are no parralels to a given line through an external point and the sum of angles and triangles is greater than 180 degrees. An understanding of this geometry can be made by considering geometry on the surface of a sphere where the shortest distance between two points is an arc of a great circle reather than a straight line. This is what seperates spherical geometry from Euclidean geometry. Because spherical geometry is based on great circles, all circles meet in two points on a spherical plane meaning that no two circles can be parralel.

Also, In spherical geometry triangles and other angles exist where great circles meet. The triangles are formed where three great circles meet, using a portion of the equator and two meridians of longitude who’s endpoints connect to one of the poles. Because the two angles at the equator would each measure 90 degrees the sum of all three would exceed 180 degrees.

Reimann geometry, which is based on the studies of german mathematician Bernhard Reimann is extremely similar to spherical geometry. Bernhard Reimann is considered to be the most influential mathematician of the mid nineteenth century. His study of geometry, called Riemann geometry or Eliptic Geometry is much like spherical geometry in the statement that there are no parralels to a given line through an external point. Reimann geometry is also known as elliptic geometry because like an elipse, a line in a plane in Reimann geometry has no point at infinity where parralels could intersect.

Very opposite the studies of Reimann, is Lobachevsky’s geometry. Lobachevsky was a russian mathematician who is not only responsible for Hyperbolic geometry but also is credited fordeveloping a method for the approximation of roots of algebraic equations. Lobachevsky’s Geometry is also known as hyperbolic geometry. It is called hyperbolic geometry because just like a hyperbola has to asymptotes, a line on a hyperbolic plane has two points at infinity. Hyperbolic geometry explores the theorum that the sum of the angles of a triangle is less than 180 degrees which contradicts Reimann, spherical, and euclidean geometry. Euclidean states that the sum of the angles of a traingle always add up to 180 and Reimann and spherical geometries both state that the sum of the angles of a triangle exceeds 180 degrees.

The best way to understand the hyperbolic plane is by using the image of a saddle-like or saddle-shaped surface. hyperbolic geometry states that two rays extending in either direction from a point, and not meeting a line are considered to be distinct parralels to that line. Another result of Lobachevsky’s geometry is the statement that there is a finite upper limit on the area of a triangle, corresponding to a traingle whose sides are parralel and all of whose angles are zero.

All four of these types of geometry were developed by brilliant mathematicians and have sufficent proof that none is more correct than the other. All of been very influencial to the world of modern math and math of their own times. They are all huge contributions to the world around and are all widley used throughout the world.