Euclidean v Non-Euclidean Geometry

The Mathematics Describing the Shape of the Universe

Feb 10, 2008

While most people are far more familiar with the basic principles of Euclidean geometry, scientists have come to discover that the shape of our universe is non-Euclidean.

Euclid of Alexandria was a 3rd century B.C. mathematician, most famous for his series of thirteen books known as the Elements, which was the most thorough description of geometrical theorems and principles that the world would see for more than two thousand years (perhaps even to this very day). Euclid began his grand work by stating five central “axioms,” and using these axioms, he proved one theorem after another, each more complex, practically inventing the principles of modern geometry.

Most of Euclid’s basic axioms are rather obvious and seem to be self-evident, such as his second axiom: “A straight line segment can be extended indefinitely in a straight line.” There are very few mathematicians who would think to dispute this, nor would they most of his other axioms.

The first problems with Euclid’s axioms did not begin to arise in a legitimate fashion until the 19th century, most notably in the works of Charles Friedrich Gauss and Janos Bolyai (who had a complex and competitive working relationship which is worth learning about in its own right) and their attempts to shed new light on the single disputed axiom of Euclid’s Elements, particularly by attempting to find a proof for Euclid's most controversial axiom.

The Parallel Postulate

Euclid’s fifth axiom, or “The Parallel Postulate” is rather simple in essence, though it has caused mathematicians no end of trouble. Simply put, the parallel postulate says that one may discover if two lines are truly parallel in the following way:

A third line is drawn which intersects both lines in question, then, by measuring the angles created by the intersection of these lines one can determine if they are parallel. Only if the two “outside” angles (and, consequently, the “inside” angles as well) created by this third line add up to exactly 180 degrees are the two lines parallel. If they add up to more or less than this, the lines are clearly going to converge at one point or another, and are therefore not parallel.

To anyone who has taken some geometry in school, this sounds on the surface to be just as reasonable as Euclid’s other axioms. So where does the trouble come from?

The Basics of Non-Euclidean Geometry

The trouble stems from the fact that Euclid’s geometry was designed to work in only specific cases – that is, on flat surfaces of two dimensions or in standard three dimensional space with straight lines. His geometry (specifically his fifth axiom) does not hold true in other circumstances.

Take, to use a classic example, a standard world globe. As the lines of longitude cross the equator, one can measure the angle thus created and, via the parallel postulate conclude that, indeed, these lines are parallel, as the angles add up to 180 degrees. Of course this is not the case, however, as these lines do, in fact, meet, both at the north and the south poles. Thus, in the case of a curved surface, one needs a non-Euclidean geometry. Another form of the same example says that on the surface of a sphere, the interior angles of a triangle need not add up to 180 degrees. In fact, a triangle on the surface of the Earth can be made using three right angles – a clear divergence from standard geometry.

The Importance of Geometries

While this all may seem rather trivial to those who do not focus their energies on mathematical studies, it surely does have great importance to the world, both in a mathematical sense, and in the realm of physical sciences.

Mathematically, non-Euclidean geometry has opened up an entirely new realm of possibilities, as such mathematicians as Gauss and Bolyai (as already mentioned) worked feverishly to develop new theorems based on curved geometrical systems.

In physics, the examples are even more clear. A fundamental feature of Einstein’s General theory of Relativity was the fact that the four dimensions of Space-Time which make up the entire universe around us are not “flat” or “smooth” as physicists had before assumed, but was rather “curved,” “twisted,” and “warped,” by the masses of objects. Thus, when developing the mathematics for his theory, he was forced to rely on non-Euclidean works in order to describe his findings. One of the titles of a chapter in Einstein’s book “Relativity: The Special and the General Theory” (Chapter 25) was called, simply, “Gaussian Co-ordinates,” and uses the work of Gauss to develop a new coordinate system for use in his theory.

In the plainest terms, non-Euclidean geometry took something that was rather simple and straightforward (Euclidean Geometry) and made it endlessly more difficult.

With this in mind, it’s sometimes hard to tell if these mathematicians should be thanked or cursed.