Non-Euclidean geometry

Euclid built geometry on five postulates. The first four are simple — for example, that any two points can be joined by a straight line. The fifth, the parallel postulate, is more complicated: if a line crosses two others and the two interior angles on one side add to less than two right angles, the two lines meet on that side if extended far enough.

For well over a thousand years, mathematicians tried to prove the fifth postulate from the first four, and failed. It took until the nineteenth century to see why: it is independent of them. Several other statements turn out to be equivalent to it, for example:

• Through a point not on a line there is exactly one parallel to it (the high-school version).
• Similar figures of any size exist: you can scale any figure up or down without changing its shape.
• There is at least one triangle whose angles add up to exactly 180°. (One such triangle is enough to make every triangle add up to 180°.)

In 1733 the Jesuit priest Girolamo Saccheri tried to prove the postulate by assuming it false and hunting for a contradiction. He never found a genuine contradiction. Guth describes him as seeing the work as a failure; historically, Saccheri published it believing he had vindicated Euclid, though his "contradiction" did not hold up.

Carl Friedrich Gauss, János Bolyai and Nikolai Lobachevsky independently took the opposite view: the fifth postulate can simply be false. In their geometry (often called hyperbolic; Guth calls it GBL geometry) there are infinitely many parallels through a point, figures of different sizes are never similar, and triangle angles add up to less than 180°. Their work appeared in the 1820s and 1830s. (In Lecture 10 Guth calls Bolyai a Prussian military man; he was a Hungarian army officer. In the Lecture 11 review Guth places the work in the later 1700s, which is too early.)

There is a second way to break the postulate, which was taken seriously only later: the surface of a sphere. There, "straight lines" are great circles; any two of them meet, so there are no parallels at all (a sphere also breaks Euclid's second postulate, since a great circle cannot be extended forever without closing on itself), and triangle angles add up to more than 180°. A triangle with one corner at the North Pole and two on the equator already has two 90° angles at the equator.

Gauss, Bolyai and Lobachevsky worked axiomatically. That left open the possibility that a contradiction lurked somewhere. The question was settled by building an explicit model with coordinates: points are pairs $(x, y)$ inside the unit disk $x^2 + y^2 < 1$, and the distance between two of them is defined by a formula Guth quotes from Felix Klein:

Here $a$ is any constant setting the length scale. The disk looks finite, but distances blow up near its edge, so it is infinite. With this distance function, all the theorems of GBL geometry hold, including the failure of the fifth postulate. So GBL geometry is as consistent as ordinary algebra with real numbers. (Guth dates Klein's paper to 1870; it is usually dated 1871, and Eugenio Beltrami had given a model of this geometry in 1868.)

Klein's model also introduced a new way of defining a geometry: not by postulates, but by a set of coordinates together with a distance function.

Gauss contributed two more ideas that general relativity is built on.

Inner properties. A surface has inner properties — everything measurable within the surface itself — and outer ones, describing how it sits in a higher-dimensional space. Gauss argued that only the inner properties matter. A two-dimensional creature living on the surface could still measure its geometry.

The metric. You do not need a formula for every long distance. Distances add up along a path, so it is enough to know the distance between infinitesimally close points. Gauss assumed that this is a quadratic expression in the coordinate differences, with coefficients that may vary from place to place:

(The factor 2 appears because $dx\,dy$ and $dy\,dx$ are the same term.) For a flat plane in Cartesian coordinates, $ds^2 = dx^2 + dy^2$. The quadratic form has an important consequence: provided it is positive (as any distance must be), every small enough patch is locally Euclidean — at any chosen point you can pick local coordinates $x', y'$ in which $ds^2 = dx'^2 + dy'^2$ there, and approximately so nearby. That is why the Earth looks flat when you only see a small piece of it. Spaces with such a metric are called Riemannian; in general relativity, where time enters with the opposite sign, they are pseudo-Riemannian.

Guth closes with why Newton's law had to be replaced. Newton's force between two masses uses their positions at the same instant, but special relativity says that "the same instant" depends on the observer. Electromagnetism had already shown the way out: a field theory, where effects travel locally at finite speed. General relativity is a field theory too — and its field is the metric of space and time itself.

• Euclid's fifth (parallel) postulate is independent of the other four; dropping it gives curved geometries.
• On a sphere there are no parallels and triangles sum to more than 180°; in GBL (hyperbolic) geometry there are infinitely many parallels and triangles sum to less.
• Klein's coordinate model proved GBL geometry consistent and introduced geometry defined by coordinates plus a distance function.
• Gauss: only inner properties matter, and a geometry is fixed by its metric $ds^2 = g_{xx}dx^2 + 2g_{xy}dx\,dy + g_{yy}dy^2$, which makes every small region locally Euclidean.