Non-Euclidean Geometry

1.1 Euclidean Geometry:
he Geometry with which we are most familiar is called Euclidean Geometry. Euclidean Geometry was named after Euclid, a Greek mathematician who lived in 300 BC. His book, called "The Elements", is a collection of postulates, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book.

Euclidean Geometry was of great practical value to the ancient Greeks as they used it (and we still use it today) to design buildings and survey land.

1.2 Spherical Geometry:
Non-Euclidean Geometry is any geometry that is different from Euclidean Geometry. One of the most useful non-Euclidean geometries is Spherical Geometry which describes the surface of a sphere. Spherical Geometry is used by pilots and ship captains as they navigate around the world. Working in Spherical Geometry has some nonintuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are South of Florida - why is flying North to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in Spherical Geometry (they lie on a "Great Circle"). Another odd property of Spherical Geometry is that the sum of the angles of a triangle is always greater then 180°. Small triangles, like ones drawn on a football field have very, very close to 180°. Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have much more then 180°.

For a more information on Spherical Geometry, see "The Geometry of the Sphere" by John C. Polking.

1.3 Hyperbolic Geometry:
The NonEuclid software is a simulation of a Non-Euclidean Geometry called Hyperbolic Geometry. Hyperbolic Geometry plays an important role in Einstein's General theory of Relativity. Hyperbolic Geometry is a "curved" space. Hyperbolic Geometry is also very important in the field of Topology. Perhaps most importantaly, Hyperbolic Geometry helps us understand what an axiematic system is: proofs, theorms, postulates, and definitions. This is the very core of all geometry, yet many students pass through geometry without grasping it.

In the next sections we will be discussing and exploring Hyperbolic Geometry in detail.

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