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 axioms, 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 non intuitive 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 significantly more than 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 is a "curved" space, and
plays an important role in Einstein's General theory of Relativity.
Hyperbolic Geometry is also has many applications within the field of Topology.

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

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Next Topic - The Shape of Space