Friday, January 1, 2010

Hyperbolic and Other Geometries (Part I)

Most people have some knowledge about regular plane, or Euclidean, geometry. The basis of many geometric theorems relies on five basic rules, or axioms, about Euclidean geometry that cannot be proved or disproved, but are simply guidelines that geometry must follow. People take these statements to be obviously true. The five are as follows (as stated by Euclid in Elements)
  1. Exactly one straight line can be drawn between any two points
  2. A finite line segment can be extended continuously without intersecting itself
  3. A circle can be made from any center and radius
  4. All right angles are equal to one another
  5. Parallel Postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if continued indefinitely, will eventually meet

In this post, the fifth of the five statements above is the one that will mainly be dealt with. This is the parallel postulate. In simple terms, as illustrated in the above image, if two lines are slightly inclined towards each other, in other words the angles between them and an intersecting line add up to less than 180 degrees, they will intersect eventually, whether it is a hundred units away, a million units away, or a googleplex units away, they will intersect. If alpha and beta in the picture above add up to more than 180 degrees, than the lines will incline away from each other and will meet only if you travel in the other direction. However, if the angles add up to exactly 180 degrees, the lines will be parallel. They will never meet. An equivalent property to this is the fact that all the angles of a triangle add up to 180 degrees. This is because if the lines are parallel, they never intersect. Yet another equivalent statement as the Parallel Postulate is the fact that for a line and a fixed point in a plane (that is a two-dimensional surface), only one line drawn through the point can be parallel to the given line (note that in three dimensions, two lines that never intersect aren't necessarily parallel. This is because they can be on two separate planes. Two lines with this relationship are called skew lines) All of the axioms above seem like common sense, right? However, most new branches of mathematics are opened up when one challenges the most simple axioms and properties of math.

In this case, we are challenging Euclid's axioms. The fifth of the statements, the Parallel Postulate, seemed the least obvious, and for centuries, various mathematicians attempted to prove this. Many of these mathematicians used a method known as proof by contradiction. Proof by contradiction begins by assuming what you're trying to prove is false and then trying to find a contradiction to a known axiom directly resulting from this fact. If a contradiction is reached, then the statement cannot be false and is therefore true. This method has be repeatedly used by mathematicians, and one notable example includes the proof that the square root of two is irrational (see here). The point is that many mathematicians assumed that the parallel postulate is false. If this is assumed to be false, then more than one line can be drawn through a point parallel to a given line. The many theorems that were derived from this assumption were very odd, indeed. However, no contradiction could be found. We now know that the Parallel postulate is neither true or false; it is true only in some cases. While searching for a contradiction, mathematicians found something much more interesting: a whole new type of geometry! In this geometry, a theorem that the angles of a triangle add up to less than 180 degrees can be derived. This new geometry is called hyperbolic geometry. It is very difficult to represent this geometry with images in Euclidean geometry, but one visual representation is shown below.

This is a representation of hyperbolic geometry in a Euclidean circle. In hyperbolic space, each angel or devil are precisely the same size. However, this representation has squeezed the entire plane of hyperbolic geometry into one circle, called the bounding circle. To the inhabitants of this Universe, you can go as far as you want, but you will never reach the bounding circle because the plane goes on forever. Shapes, lines, and other Euclidean structures can all be represented in hyperbolic geometry, although they may look different than in Euclidean space. Here is a triangle as it looks like on the hyperbolic plane.

The values shown are of each angle along with the total sum off to the right. Note the the sum of the angles, 107.1 degrees, is much smaller than what it "should" be, 180. In fact, it was discovered that the value of 180 minus the sum of the three angles is proportional to the area of the triangle, a relation somewhat unexpected by mathematicians. The following equation represents this relation.


x, y, and z represent the three angles of a triangle and the symbol Δ represents the area of the triangle. In this equation, the constant C is not a set value for all hyperbolic planes. In fact, unlike Euclidean geometry, there are different types of hyperbolic planes, which will be mentioned later. Note that if the line segments on the above triangle were continued into lines, they would continue curving and also "reach" the bounding circle (obviously the finite curve that we see intersecting the bounding circle is indeed infinite from a hyperbolic standpoint).

Returning to the origin of this geometry, is the parallel postulate indeed false on this plane? Is is possible to draw more than one parallel line through a given point? Yes. The picture below is a different visual representation of hyperbolic geometry, but the structures are similar.

This representation may look different, but it is viewing the same plane. The lines in the lower right are parallel, but they curve in opposite directions, rather than remaining a fixed distance fro m each other. From this image, it is clear that infinitely many lines could be drawn through the same point as the given line, but would still never intersect it. Euclid's parallel postulate is indeed false in this geometry.

For the next part of this post, see Hyperbolic and Other Geometries (Part II).

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