Philosophy of Maths Non-Euclidean Geometries

Two lines are parallel if they “go in the same direction”. In Euclidean geometry, we often say they’re parallel if they never meet, even if we extend them “forever” (whatever that means – and note this is not how Euclid puts it in the 5 th postulate). Another way to say this is that they stay the same distance apart, just as railway tracks do. For now we’ll stick with “straight” lines. A line is straight if it’s the shortest distance between two points. We’ll see later that this idea isn’t straightforward.

Suppose we have the line CD and some other point E. We’d like a parallel line to CD through E (say, to make them a pair of railway tracks). It seems there’s only one choice that works (in red). Euclid effectively asserts this as his famous “Fifth Postulate”. For a long time geometers tried to prove the Fifth Postulate.

In the nineteenth century, though, new geometries were discovered (or invented?) that are exactly the same as Euclid’s but where the Fifth Postulate is false. In elliptic space there are no parallel lines: every line through x must meet a somewhere. Meanwhile, in hyperbolic space there are an infinite number of lines through x that never meet a! To make sense of this we’ll need to adjust our intuition about what a “straight line” is. This showed that the Fifth Postulate can never be proved from the other axioms (postulates, definitions and common notions).

Straight Lines Euclid’s “definition” of a straight line is not much help: – “A straight line lies equally with respect to the points on itself”. – This is so useless that some scholars even think it was added by a later editor, influenced by the much later Definitions of terms in geometry attributed to Hero of Alexander (c.50AD). Our modern definition is usually “the shortest path between two points”, but it’s not known who first came up with this formulation. – On a flat page, the shortest distance between two points is indeed the one you get by connecting them with a straightedge, per Euclid’s Proposition 1. – On a sphere, though, your straightedge doesn’t work so well. – It was probably the Sphaerica of Menelaus, c100AD, that first pointed out that we can make sense of the idea of a “straight line” between two points on the sphere if we think in terms of shortest distance.

ELLIPTIC GEOMETRY

Renaissance armillary sphere

Persian celestial globe, c1144

Straight Lines Walking in a straight line on the surface of the Earth means walking along a “great circle”

So elliptic geometry is Euclidean geometry but with the 5 th postulate replaced by the following statement: Given a line a and a point x not on a, there are no lines through x that do not intersect a.

The Biangle Walking in a straight line on the surface of the Earth means walking along a “great circle”

Triangles in spherical geometry are “fat”: their angles add up to more than 180 ○. This is easiest to see with a division of the sphere into eight equilateral triangles; each angle must be 90 ○, making a total angle sum of 270 ○ !

HYPERBOLIC GEOMETRY

Carl Friedrich Gauss ( ) was one of the most prominent mathematicians of the period we’re looking at. He described a simple and intuitive way to measure how curved a surface is at a point. The curvature of a straight line at a point on a surface can be thought of as a measure of how hard it would be to walk along that line (i.e. how uphill it is). The Gaussian curvature at a point on a surface is calculated like this: 1.Stand at the requisite point. 2.Look in a straight line (i.e. along a geodesic!) in a direction where the curvature is least: that is, where it’s easiest to walk. 3.Do the same in the direction where the curvature is greatest: that is, where it’s hardest to walk. 4.Multiply the two together to get Gaussian curvature.

The plane, where we do Euclidean geometry, has constant Gaussian curvature 0. This is because every line has 0 curvature at every point, and 0x0=0. The sphere, where we do elliptic geometry, has constant Gaussian curvature 1. Every way you travel on a sphere is “downhill” in exactly the same way. Spaces of constant Gaussian curvature “look and feel” the same, wherever you are in them.

A Pringle crisp doesn’t have a constant Gaussian curvature but if you were an ant standing in the centre of it you would get to experience negative curvature. This is because in one direction the surface curves downwards, and in the other it curves upwards. Gaussian curvature is calculated by multiplying these two “principal curvatures” together. Since one must be positive and the other negative, the end result must be negative.

Positive Gaussian curvature gives triangles whose angles sum to more than 180° Triangles on a surface of zero Gaussian curvature have angles summing to exactly 180° Negative Gaussian curvature gives triangles whose angles sum to less than 180°

You can experience negative curvature if you live on a torus, too, but you have to be somewhere on the “inside” of the hole. On the “outside” everything curves away from you in the same way (you feel like you’re on top of a hill all the time) so the curvature must be positive there. The curvature of a space affects its geometry, so the torus is an inconvenient place to study geometry because what’s true in one place might not be true in another. It’s much easier to do geometry in spaces of constant curvature.

The angles in a triangle change when you slide it around on a torus – the geometry in one place is different from in the other.

Discovery of Hyperbolic Geometry Is there a space in which the Gaussian curvature is a constant negative value? – Philosophically, this question takes us back to Plato and Mill: what are we really asking here? – The difficulty is that we can’t imagine such a space: when we try, it ends up folding back on itself and “running out of room”. We can fit a 2D Euclidean plane in a 2D space, like a flat tabletop. We need a 3D space to fit a sphere, even though its surface is only 2D. For a 2D “hyperbolic plane” we would need at least 4 dimensions! In the 1820s Janos Bolyai explored this question and constructed a whole theory of geometry in such a space. – He wrote to his father, who had warned him against pursuing this topic, “Out of nothing I have created a strange new universe”. At the same time, Nicolai Lobachevsky was working independently on a geometry in which many parallels exist through a point. – Maybe you could say it was an idea whose time had come, but both men died in poverty and having won no recognition for their work. – It was only with Einstein’s theory of Special Relativity that hyperbolic space found a physical application: in a sense, SR suggests that spacetime itself is hyperbolic, not Euclidean.

Where Does This Leave Us? Kant was able to believe that Euclidean geometry is the form of outer intuition in part because it appeared to be the only contender. – Although spherical geometry would have been well-known to him, it was presented as a special set of rules for astronomy or navigation, not an alternative geometry. It was obvious that the universe is really Euclidean. – “A geometry” wouldn’t even make sense: there was just “geometry”. When this changed, geometry was relativised. We now had to choose which geometry to use, and it seemed we could even invent fanciful ones that have no physical existence. – This raises (yet again!) two central questions: what does the reality of geometric objects consist in, and do we discover or invent them?