[Math Lair] Euclidean and Non-Euclidean Geometry

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Euclid's Elements begins, following several definitions, with five postulates. These five postulates define Euclidean geometry. They are:

  1. To draw a straight line from any point to any point.
  2. To produce a finite straight line continuously in a straight line.
  3. To describe a circle with any centre and distance.
  4. That all right angles are equal to one another.
  5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

It is obvious, by just looking at the number of words in each postulate, that the fifth postulate is somewhat different from the other four. Also unlike the other four postulates, it isn't used until Proposition 29 in Book I.

There are several other ways of stating the fifth postulate. Two of the best-known were discovered by Proclus in the fifth century A.D. They are:

Through a given point only one parallel can be drawn to a given straight line or Two straight lines which intersect one another cannot both be parallel to one and the same straight line.

(this is sometimes known as Playfair's postulate after the nineteenth-century English mathematician John Playfair) and

If a straight line intersect one of two parallels, it will intersect the other also.

(this is known as Proclus' axiom).

Due to the complexity of the wording of the fifth postulate, you might wonder whether it is really needed or not, or whether it could be deduced from the other four. Actually, it is needed. It is a testimony to Euclid's genius that he saw that this postulate was required. However, this genius was not recognised until relatively recently. Many people throughout the ages had tried to show that the fifth postulate is really a theorem that could be deduced from the other four. The first attempt that we know of was made by Posidonius in the first century B.C. He used a definition of parallel lines that was different from Euclid's. Posidonius' definition of parallel lines, in contrast to that of Euclid's which stated that they never meet, was that parallel lines were lines the distance between which is always the same. This definition assumes more than Euclid's definition. Many other attempts were made over the next two millennia to prove the fifth postulate from the other four.

There is little evidence that anyone thought that the fifth postulate could not somehow be demonstrated from the other four until 1763, when Georg Klügel finished a doctoral dissertation that examined the literature on the fifth postulate. Klügel examined 28 attempts to prove the postulate, found them all insufficient, and suggested that the postulate could not be proven, and the fact that we believe it to be true is because of experimental observation.

One of the most important parts of mathematical problem solving is determining the right question to ask. Following Klügel's work, other mathematicians started to consider the implications of the fifth postulate being false. Gauss saw that Euclid's geometry was not the only possible geometry, considering such a geometry but never publishing his work during his lifetime. After his work was published, it was received with interest and also ignited interest in work previously published by Nikolai Lobachevski and János Bolyai. These works laid the foundations for two new types of geometry, hyperbolic geometry and elliptical geometry.

Elliptical geometry can be thought of as the geometry of a surface of a sphere. Parallel lines meet up with each other. For example take, on the surface of the Earth, lines of longitude, which are parallel with each other, meet at the North and South Poles. The three angles of a triangle drawn on the surface of the Earth will sum to more than 180°. If you draw a fairly small triangle, the difference won't be noticeable, but if you were to draw a triangle from the North Pole along the Greenwich Meridian to the equator, from there along the equator to a point at 90° west latitude, and back to the north pole, this triangle would have three 90° angles, for a total of 270°.

Hyperbolic geometry could be thought of as the geometry of a surface similar to the inner surface of a bowl. On this surface, parallel lines, instead of remaining the same distance from one another as in Euclidean geometry or converging as they do in elliptical geometry, get further and further away from each other the further you go. Triangles have angles less than 180°.

There are several applications of non-Euclidean geometry; one is that it helps describe the picture of the universe painted by Einstein's Theory of Relativity. Space is not perfectly "flat", or Euclidean; the force of gravity gives it a curvature, especially in the vicinity of massive objects.