There are certain geometrical principles which are often taken for granted by those who have learned the subject over the course of several years of schooling.
Such principles include:
- The inside angles of every triangle add up to exactly 180 degrees.
- Parallel lines never meet.
and
- A straight line will never meet up with itself.
These are very basic, intuitive, and seemingly self-evident principles - some of the most basic which arose out of Euclid's Elements, the first foundational theory of mathematical geometry.
During the 19th century and beyond, however, mathematicians began to realize that Euclid's wasn't the only geometry available, and the rise of non-Euclidean alternatives demonstrated two key alternatives: Hyperbolic geometry, and Spherical geometry. The former is that which is defined by a "saddle shaped" curvature. The latter, however, is more self-evident: Spherical geometry is the geometry which best describes, what else, the surface of a sphere.
Non-Euclidean Triangles
Each of those basic Euclidean principles listed above are thrown out the window in spherical geometry, no matter how obvious they may have seemed.
Perhaps the best tool for visualizing this form of geometry is by imagining something most people are quite familiar with: a globe. A simple spherical map of the Earth.
First, in Euclidean geometry the inside angles of every triangle add up to exactly 180 degrees, which is really quite convenient. In spherical geometry, however, every triangle has angles which must add up to more than 180 degrees. In fact, one can imagine a triangle even with angles of 270 degrees or more!
Imagine drawing a line on a globe, beginning at the north pole, traveling south to the equator, turning ninety degrees in either direction and traveling 1/4 of the way around the world, then turning 90 degrees again and heading back to the equator. The result? A triangle made entirely of right triangles - something which to many should seem fundamentally impossible.
Parallel Lines
Similarly, anyone familiar with world maps knows that the vertical lines (lines of longitude) all meet up with each other at both the north and south poles, even though they are running exactly parallel at the equator! In fact, in spherical geometry there are no parallel lines, for every line must meet up somewhere.
What, one may ask, about the lines of latitude which traverse the world without meeting each other? Technically, these are not considered straight lines, as they do not follow a "geodesic," which may be defined as "the straightest possible path through curved space." Lines of latitude actually could be considered the spherical equivalent of "curved lines," which is necessary to keep them from colliding with each other.
In spherical geometry, then, one requires an entirely new system of geometry. Fortunately, at very small scales, at specific locations, as every person witnesses on a daily basis, even a fundamentally non-Euclidean surface such as the Earth will "behave" as if it is perfectly flat and Euclidean. It is only once one begins to look at the big picture - whether it is the surface of the Earth or the entire universe as a whole - that the non-Euclidean nature of things begins to come to light.
Isaac M. McPhee - Isaac McPhee was born as a human child in Mt. Vernon, WA, c. 1982; he currently resides in the bustling heart of New York City where he ...
