# Geometry

### Euclidean Geometry

Euclidean geometry is the study of points, lines, planes, and other geometric figures, using a modified version of the assumptions of Euclid (c.300 BC). The most controversial assumption has been the parallel postulate: there is one and only one line that contains a given point and is parallel to a given line. The development of Euclidean geometry extends at least from 10,000 BC to the 20th century.

In the 4th century BC, Plato founded an Academy in Athens, emphasized geometry, and used the five regular Polyhedrons in his explanation of the scientific phenomena of the universe. Aristotle, a student of Plato at the Academy, identified the rules for logical reasoning. The 13 books of Euclid's Elements are based on the mathematics that was considered at Plato's Academy.

The geometry in the Elements was a logical system based on ten assumptions. Five of the assumptions were called common notions (Axioms, or self-evident truths), and the other five were postulates (required conditions). The resulting logical system was taken as a model for deductive reasoning and had a profound effect on all branches of knowledge. Although it has been necessary to refine the postulates as concepts of existence, continuity, order, and other aspects of Geometry have changed, the resulting geometry is still called Euclidean geometry.

Modifications of Euclid's parallel postulate provide the basis for non-Euclidean Geometry. The currently accepted set of postulates for Euclidean geometry was first proposed in 1899 by David Hilbert.

Euclid's five postulates on which he based all his theorems:

1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

From Euclid's ideas in Geometry came G.D. Birkhoffs notions of Geometry:

1. Two points determine a line
2. Lines go on indefinitely and lengths add
3. Lines separate the plane into two convex parts
4. Angles make sense and sum to 180 degrees, yielding a straight line
5. "Side Angle Side"

It was Euclid's Fifth postulate (Parallel Postulate) which showed the greatest challenge in proving.

The ideas of the following Mathematicians contributed to the proof of this postulate:

Kayyan (1050-1123)
Vitali (1633-1711)
Saccheri (1667-1733)
Playfair (1748-1819)

The following Mathematicians also went on to "prove" a geometry which follows only the first 4 postulates (Hyperbolic Geometry):

Gauss (1816-1831)
Bolzai (1823)
Lobachevsky (1829-1830)
Beltrami (1868)

Further development also included the Poincare Disk

Well known application to the Poincare disk is MC Eschers work with tesselations on the Poincare:

As can be seen from the developments, Euclidean Geometry (geometry on the plane) was used to develop Non-Euclidean geometry (geometry on the suface of a sphere) and also hyperbolic geometry.

This site was designed by Jacqueline Cordero and Theresa Fulton.