Geometry is one of the oldest branches of mathematics, and most important among texts is Euclid's Elements. His text begins with 23 definitions, 5 postulates, and 5 common notions. From there Euclid starts proving results about geometry using a rigorous logical method, and many of us have been asked to do the same in high school.

Euclid's Elements served as the text on geometry for over 2000 years, and it has been admired as a brilliant work in logical reasoning. But one of Euclid's five postulates was also the center of a hot debate. It was this debate that ultimately led to the non-Euclidean geometries that can be applied to different surfaces.

Here are Euclid's five postulates:

One can draw a straight line from any point to any point.

One can produce a finite straight line continuously in a straight line.

One can describe a circle with any center and radius.

All right angles equal one another.

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 produced indefinitely, meet on that side on which the angles are less than two right angles.

Does one postulate not look like the others? The first four postulates are short, simple, and intuitive. Well, the second might seem a bit odd, but all Euclid is saying here is that you can produce a line segment to any length you want. However, the 5th one, called the parallel postulate, is not short or simple; it sounds more like something you would try to prove than something you would take as given.

Indeed, the parallel postulate immediately gave philosophers and other thinkers fits, and many tried to prove that the fifth postulate followed from the first four, to no avail. Euclid himself may have been bothered at some level by the parallel postulate since he avoids using it until the proof of the 29th proposition in his text.

In trying to make sense of the parallel postulate, many equivalent statements emerged. The two equivalent statements most relevant to our study are these:

5\(^\prime\text{.}\) Given a line and a point not on the line, there is exactly one line through the point that does not intersect the given line.

\(5^{\prime\prime}\text{.}\) The sum of the angles of any triangle is 180\(^\circ\text{.}\)

Reformulation \(5^\prime\) of the parallel postulate is called Playfair's Axiom after the Scottish mathematician John Playfair (1748-1819). This version of the fifth postulate will be the one we alter in order to produce non-Euclidean geometry.

The parallel postulate debate came to a head in the early 19th century. Farkas Bolyai (1775-1856) of Hungary spent much of his life on the problem of trying to prove the parallel postulate from the other four. He failed, and he fretted when his son János (1802-1860) started following down the same tormented path. In an oft-quoted letter, the father begged the son to end the obsession:

For God's sake, I beseech you, give it up. Fear it no less than the sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life.^{ 1 }See for instance, Martin Gardner's book The Colossal Book of Mathematics, W.W. Norton & Company (2001), page 176.

But János continued to work on the problem, as did the Russian mathematician Nikolai Lobachevsky (1792-1856). They independently discovered that a well-defined geometry is possible in which the first four postulates hold, but the fifth doesn't. In particular, they demonstrated that the fifth postulate is not a necessary consequence of the first four.

In this text we will study two types of non-Euclidean geometry. The first type is called hyperbolic geometry, and is the geometry that Bolyai and Lobachevsky discovered. (The great Carl Friedrich Gauss (1777-1855) had also discovered this geometry; however, he did not publish his work because he feared it would be too controversial for the establishment.) In hyperbolic geometry, Euclid's fifth postulate is replaced by this:

5H. Given a line and a point not on the line, there are at least two lines through the point that do not intersect the given line.

In hyperbolic geometry, the sum of the angles of any triangle is less than 180\(^\circ\text{,}\) a fact we prove in Chapter 5.

The second type of non-Euclidean geometry in this text is called elliptic geometry, which models geometry on the sphere. In this geometry, Euclid's fifth postulate is replaced by this:

5E. Given a line and a point not on the line, there are zero lines through the point that do not intersect the given line.

In elliptic geometry, the sum of the angles of any triangle is greater than 180\(^\circ\text{,}\) a fact we prove in Chapter 6.

The Pythagorean Theorem

The celebrated Pythagorean theorem depends upon the parallel postulate, so it is a theorem of Euclidean geometry. However, we will encounter non-Euclidean variations of this theorem in Chapters 5 and 6, and present a unified Pythagorean theorem in Chapter 7, with Theorem 7.4.7, a result that appeared recently in [20].

The Pythagorean theorem appears as Proposition 47 at the end of Book I of Euclid's Elements, and we present Euclid's proof below. The Pythagorean theorem is fundamental to the systems of measurement we utilize in this text, in both Euclidean and non-Eucidean geometries. We also remark that the final proposition of Book I, Proposition 48, gives the converse that builders use: If we measure the legs of a triangle and find that \(c^2 = a^2 + b^2\) then the angle opposite \(c\) is right. The interested reader can find an online version of Euclid's Elements here [29].

Theorem1.2.1The Pythagorean Theorem

In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.

Suppose we have right triangle \(ABC\) as in Figure 1.2.2 with right angle at \(C\text{,}\) and side lengths \(a, b\text{,}\) and \(c\text{,}\) opposite corners \(A, B\) and \(C\text{,}\) respectively. In the figure we have extended squares from each leg of the triangle, and labeled various corners. We have also constructed the line through \(C\) parallel to \(AD\text{,}\) and let \(L\) and \(M\) denote the points of intersection of this line with \(AB\) and \(DE\text{,}\) respectively. One can check that \(\Delta KAB\) is congruent to \(\Delta CAD\text{.}\) Moreover, the area of \(\Delta KAB \) is one half the area of the square \(AH\text{.}\) This is the case because they have equal base (segment \(KA\)) and equal altitude (segment \(AC\)). By a similar argument, the area of \(\Delta DAC\) is one half the area of the parallelogram \(AM\text{.}\) This means that square \(AH\) and parallelogram \(AM\) have equal areas, the value of which is \(b^2\text{.}\)

One may proceed as above to argue that the areas of square \(BG\) and parallelogram \(BM\) are also equal, with value \(a^2\text{.}\) Since the area of square \(BD\text{,}\) which equals \(c^2\text{,}\) is the sum of the two parallelogram areas, it follows that \(a^2+b^2 = c^2.\)

The arrival of non-Euclidean geometry soon caused a stir in circles outside the mathematics community. Fyodor Dostoevsky thought non-Euclidean geometry was interesting enough to include in The Brothers Karamazov, first published in 1880. Early in the novel two of the brothers, Ivan and Alyosha, get reacquainted at a tavern. Ivan discourages his younger brother from thinking about whether God exists, arguing that if one cannot fathom non-Euclidean geometry, then one has no hope of understanding questions about God.^{ 2 }See, for instance, The Brothers Karamazov, Fyodor Dostoevsky (a new translation by Richard Pevear and Larissa Volokhonsky), North Point Press (1990), page 235.

One of the first challenges of non-Euclidean geometry was to determine its logical consistency. By changing Euclid's parallel postulate, was a system created that led to contradictory theorems? In 1868, the Italian mathematician Enrico Beltrami (1835-1900) showed that the new non-Euclidean geometry could be constructed within the Euclidean plane so that, as long as Euclidean geometry was consistent, non-Euclidean geometry would be consistent as well. Non-Euclidean geometry was thus placed on solid ground.

This text does not develop geometry as Euclid and Bolyai and Lobachevsky did. Instead, we will approach the subject as the German mathematician Felix Klein (1849-1925) did.

Whereas Euclid's approach to geometry was additive (he started with basic definitions and axioms and proceeded to build a sequence of results depending on previous ones), Klein's approach was subtractive. He started with a space and a group of allowable transformations of that space. He then threw out all concepts that did not remain unchanged under these transformations. Geometry, to Klein, is the study of objects and functions that remain unchanged under allowable transformations.

Klein's approach to geometry, called the Erlangen Program after the university at which he worked at the time, has the benefit that all three geometries (Euclidean, hyperbolic and elliptic) emerge as special cases from a general space and a general set of transformations.

The next three chapters will be devoted to making sense of and working through the preceding two paragraphs.

Like so much of mathematics, the development of non-Euclidean geometry anticipated applications. Albert Einstein's theory of special relativity illustrates the power of Klein's approach to geometry. Special relativity, says Einstein, is derived from the notion that the laws of nature are invariant with respect to Lorentz transformations.^{ 3 }Relativity: The Special and General Theory, Crown Publications Inc (1961), p. 148.

Even with non-Euclidean geometry in hand, Euclidean geometry remains central to modern mathematics because it is an excellent model for our local geometry. The angles of a triangle drawn on this paper do add up to 180\(^\circ\text{.}\) Even “galactic” triangles determined by the positions of three nearby stars have angle sum indistinguishable from 180\(^\circ\text{.}\)

However, on a larger scale, things might be different.

Maybe we live in a universe that looks flat (i.e., Euclidean) on smallish scales but is curved globally. This is not so hard to believe. A bug living in a field on the surface of the Earth might reasonably conclude he is living on an infinite plane. The bug cannot sense the fact that his flat, visible world is just a small patch of a curved surface (Earth) living in three-dimensional space. Likewise, our apparently Euclidean three-dimensional universe might be curving in some unseen fourth dimension so that the global geometry of the universe might be non-Euclidean.

Under reasonable assumptions about space, hyperbolic, elliptic, and Euclidean geometry are the only three possibilities for the global geometry of our universe. Researchers have spent significant time poring over cosmological data in hopes of deciding which geometry is ours. Deducing the geometry of the universe can tell us much about the shape of the universe and perhaps whether it is finite. If the universe is elliptic, then it must be finite in volume. If it is Euclidean or hyperbolic, then it can be either finite or infinite. Moreover, each geometry type corresponds to a class of possible shapes. And, if that isn't exciting enough, the overall geometry of the universe may be fundamentally connected to the fate of the universe. Clearly there is no more grand application of geometry than to the fate of the universe!

Use Euclid's parallel postulate to prove the alternate interior angles theorem. That is, in Figure 1.2.3(a), assume the line \(BD\) is parallel to the line \(AC\text{.}\) Prove that \(\angle BAC = \angle ABD\text{.}\)

2

Use Euclid's parallel postulate and the previous problem to prove that the sum of the angles of any triangle is 180\(^\circ\text{.}\) You may find Figure 1.2.3(b) helpful, where segment \(CD\) is parallel to segment \(AB\text{.}\)