Non-Euclidean geometry

What is a point? What is a line?

It all began with Euclid’s effort to gather geometric knowledge into a single work, the Elements (around 300 BCE; the link leads to a Czech translation).

Euclid tried to define the basic objects, but from today’s point of view his definitions were rather intuitive and philosophical:

• he defined a point as “that which has no part”,
• and a line as “breadthless length”.

To build his theory, he laid down five postulates:

1. Any two points can be joined by a straight line. try it
2. A line segment can be extended indefinitely. try it
3. A circle can be drawn with any centre and any radius. try it
4. All right angles are equal to one another. try it
5. If a straight line crosses two other straight lines so that the interior angles on the same side add up to less than two right angles, then the two lines (if extended far enough) meet on that side. try it

While the first four were considered obvious and simple, the fifth postulate (about parallels) stood out for its complexity. Euclid himself avoided it as long as he could and used it for the first time only in Proposition 29.

In the Euclidean plane, the interior angles of every triangle add up to exactly 180°. Try it yourself; at the next stops we will compare this with the sphere and with the hyperbolic plane.

Is the fifth postulate really an independent assumption, or can it be proved from the first four?

For centuries, mathematicians believed that it was not an axiom but a theorem that could be proved from the preceding four.

For two thousand years, a long line of mathematicians tried to find a proof: Ptolemy, Proclus, Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, John Wallis, Girolamo Saccheri, Johann Heinrich Lambert, Adrien-Marie Legendre and many others. None of the “proofs” held up. Each time it turned out that the author had quietly used a statement equivalent to the fifth postulate.

These attempts did, however, produce a number of equivalent formulations. The two best known are:

Assuming the first four postulates, these statements are logically equivalent to the fifth postulate: denying one necessarily means refuting the other. The essential assumption, however, is the Euclidean plane. What happens if we leave it and carry our reasoning over to, say, the surface of a sphere?

What if the “plane” is not a flat sheet of paper but the surface of a sphere?

Model. The “plane” is the surface of a sphere, and the “points” are points on it. The role of “lines” is taken over by great circles (link in Czech), i.e. the intersections of the sphere with planes passing through its centre (meridians, the equator). They are the shortest paths between points along the surface of the sphere.

The postulate fails. Any two great circles intersect, and they even do so at two antipodal points. So parallels do not exist at all.

The triangle. It is easy to imagine a triangle formed by two meridians and a stretch of the equator. The meridians meet the equator at an angle of 90°, so

α + β + γ = 90° + 90° + φ > 180°,

where φ is the angle between the meridians at the pole. For φ = 90° we get a triangle with three right angles.

In general, the larger the triangle, the more its angle sum exceeds 180°:

α + β + γ − 180°  is proportional to the area of the triangle.

Small triangles have an angle sum of almost exactly 180°, which is why the Earth’s surface looks flat to us on an everyday scale. This simple example shows that without the fifth postulate the whole Euclidean theory of triangles collapses too.

What if, through a point not on a line, there are neither zero parallels nor exactly one, but infinitely many?

While spherical geometry is a model of a space with positive curvature, hyperbolic geometry is its opposite, i.e. a space with constant negative curvature. You can picture it as a surface that is “curved like a saddle” everywhere.

The postulate fails. The fifth postulate is replaced by the opposite statement: through a point not on a line there pass infinitely many lines that do not meet the given line.

Model: the Poincaré disc.

• The “plane” is the interior of a circle. Its boundary lies “at infinity”.
• The “lines” are the diameters of the circle and arcs of circles that meet the boundary at right angles.
• Angles are measured as usual, but distances “stretch” towards the boundary. An inhabitant of the disc never reaches the boundary.

The triangle. The sides of a hyperbolic triangle are “bent inwards”, and so:

• the sum of the interior angles is always less than 180°,
• the difference 180° − (α + β + γ) (the so-called angular defect) is proportional to the area: the larger the triangle, the smaller its angle sum,
• a triangle with its vertices “at infinity” (on the boundary of the disc) has all its angles equal to zero, and yet a finite area.

What a hyperbolic plane tiled with tiles of equal size looks like is shown by M. C. Escher’s woodcuts, for example Circle Limit III (1959). The tiles near the boundary look tiny, but in hyperbolic geometry they are all the same size.

Who was the first to realise that the fifth postulate cannot be proved?

Around 1820, three men independently reached the same conclusion: the fifth postulate cannot be proved, because there is a consistent geometry in which it does not hold. Yet their fates could not have been more different.

Carl Friedrich Gauss (1777–1855): a discovery in a drawer

The “Prince of Mathematicians” from Göttingen was already thinking about parallels as a student. By around 1820 he was convinced that non-Euclidean geometry was consistent. Yet he never published anything about it. In a letter to the astronomer Bessel in 1829 he explained why: he feared the “clamour of the Boeotians”, that is, the mockery of those who would not understand the new idea. Gauss was famous and cautious and did not want to risk his reputation. He confided his thoughts only to letters and private notes. It is said that during his geodetic survey of Hanover he also examined the angle sum of a large triangle with the vertices Hohenhagen, Brocken and Inselsberg. Whether he really wanted to “test” non-Euclidean geometry this way, however, is more of a legend.

János Bolyai (1802–1860): a father’s warning

János’s father, Farkas Bolyai, a friend of Gauss from their student days, devoted a large part of his life to parallels and solved nothing. When he found out that his son was working on them too, he begged him in a letter to give it up: it would take away his peace, his health and his joy in life, just as it had done to him. János, a young officer in the Austrian army, carried on all the same. In 1823 he wrote to his father with enthusiasm that he had “created a new, strange world out of nothing”. His work appeared in 1832 only as a supplement (the Appendix) to his father’s textbook. Farkas sent it to Gauss, who replied that he could not praise it, because that would mean praising himself: he said he had arrived at the same thing years before. For János it was a tremendous blow. He felt that Gauss wanted to deny him the discovery. When he later learned about Lobachevsky as well, he suspected that he was a made-up figure. He left the army, lived in seclusion and poverty in Marosvásárhely (today Târgu Mureș in Romania) and never published anything again. He died in 1860, almost forgotten. After his death he left thousands of pages of manuscripts.

Nikolai Ivanovich Lobachevsky (1792–1856): ridicule in Kazan

Lobachevsky was a professor and for many years rector of the University of Kazan. He presented his “imaginary geometry” in 1826 and published it in 1829–1830 in the local university journal, the first of the three to do so. The response was harsh. The influential St Petersburg mathematician Mikhail Ostrogradsky rejected the work, and a mocking anonymous review appeared in a St Petersburg magazine. Lobachevsky did not give up and kept writing, in Russian, German and French. Gauss read his little German book of 1840, reportedly began learning Russian because of Lobachevsky’s works, and pushed through his election to the Göttingen scientific society (1842). But he never stood up for him in public, and Lobachevsky himself probably never learned of this recognition. In 1846 Lobachevsky was removed from the university. Towards the end of his life he went blind and had to dictate his last work, Pangeometry (1855). He died in 1856 without having received recognition.

Late vindication. Only after all three had died did the situation turn around. Gauss’s posthumous correspondence revealed that he had believed in the new geometry. Eugenio Beltrami (1868) and later Felix Klein and Henri Poincaré constructed models of hyperbolic geometry inside Euclidean geometry. In this way they showed once and for all that it is just as consistent as Euclid’s. The “proof” of the fifth postulate sought for two thousand years therefore does not exist. The postulate is independent of the others. Today hyperbolic geometry bears Lobachevsky’s name, and in Hungary and Romania Bolyai is a national hero.

Why should we care about geometries in which the fifth postulate does not hold?

The discovery of non-Euclidean geometries changed the way we look at mathematics itself.

Axioms stopped being “self-evident truths about the world” and became a choice. A mathematician can choose them differently and explore what kind of world grows out of them. All of modern axiomatic mathematics rests on this insight.

The different geometries also have very practical uses. Spherical geometry is the basis of navigation and cartography: aircraft and ships travel along great circles, and surveyors measuring large territories have to take the spherical excess into account.

Hyperbolic geometry has found its way into the visual arts (Escher’s “circle limits”) and in modern computer science it is used to display huge networks and hierarchical data. And it opened a question that Euclid never asked: what geometry does the space we actually live in have?