CH 8: In Addition to High School Geometry
For those of you who have made it through high school geometry, terms like π (Chapter 17), acute, obtuse, parallel and perpendicular probably sound familiar. One of the central tenants of high school geometry states that the angle sum of any triangle, also known as a 3-gon, must always equal 180 degrees. The truth is that in many situations these so called triangles can actually have more or less than 180 degrees. This is due to the fact that high school geometry does not always explain that there are different types of geometric systems.
High school geometry is formally called Euclidean geometry and refers to the geometry we see with modern architecture, and most man-made creations. In nature, however, geometry doesn't always follow such a precise script. For most of human history, it was assumed that Euclidean Geometry was absolute; there was nothing else to even consider. However, in the past few centuries, we have come to realize there are other geometries that can accurately describe physical space. Two of the most common Non-Euclidean types are called Hyperbolic and Elliptic Geometries. The way in which these geometries differ is in the modification of a central tenant of Euclidean geometry, the Parallel Postulate. The reality is that in other geometries parallel lines simply do not exist. Part of this reason is that Euclidean geometry is modeled on the notion of a flat plane. For example, when you consider the Earth, a sphere, we have another geometry to describe its surface, Elliptic, and where the sum of the angles of a triangle on such a surface will exceed 180 degrees.
The 3rd Rock from the Sun is Non-Euclidean