by Dean Traylor

Created on: January 21, 2011   Last Updated: January 24, 2011

In the evolution of math, established beliefs and understanding of analytical thinking will always be challenged. And, in many cases, those new ways of thinking about math will eventually replace, modify, or give birth to a new way of understand this discipline.

Non-Euclidean geometry is one example. This concept is aptly named, for it refers to geometry that doesn’t adhere to all the postulates of Euclidean spaces (or to be more precise, any concept of geometric shapes and spaces that doesn’t follow traditional views held since Euclid’s time).

This specification is a term that refers to several theories of geometry: Hyperbolic geometry (also known as Lobachevsky-Bolyai-Gaus geometry), elliptic geometry (or Riemannian geometry), and Spherical geometry.

The difference between Euclid geometry and non-Euclidean geometry can be characterized by the dimensions used. According to the website, Wolfram Math World, Euclidean geometry, (also known as parabolic geometry,) is  a “flat” geometry; it focuses on one dimensional shapes or spaces . Non-Euclidean geometry explores two and three dimensions within a shape.

Euclid’s influence on geometry is undeniable. In 300 BC, the Greek mathematician of antiquity wrote “Elements,” the most influential text on geometry. His concepts would become the hallmark of math and science. It would also influence countless mathematicians, physicists, astronomers, and other scientists for thousands of years.

In his famous book, Euclid described five postulates (According to Encarta Dictionary, a statement that is assumed to be true but has not been proven and that is taken as the basis for a theory, line of reasoning, or hypothesis.) His theorems for these postulates are as follows:

1. To draw a straight line from any point to any other

2. To produce a finite straight line continuously in a straight line continuously in a straight line

3. To describe a circle with any center and distance

4. All right angles are equal to each other

5. If a straight line falling on two straight lines making 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 (O’Conner, Robertson: 1996).

According J.J. O’Conner and E.F. Robertson, of St. Andrew Academy of the United Kingdom, the fifth postulate is “different from the other four. It did not satisfy Euclid and he tried to avoid its

• 1
• 2
• 3
• Next Page >>