The nineteenth century was a tumultuous time to be a geometer. The world of triangles, squares and circles, the immortal, perfect figures studied since ancient times, was in turmoil. Philosophers and mathematicians had discovered that the simple rules of shapes—that the angles inside a triangle always add up to 180°, that circumference of a circle is 2πr—could be broken.
In fact, these rules are very fragile—try to do mathematics somewhere only slightly unconventional, like on the surface of a sphere, and ordinary geometry breaks down. If you are standing at the North Pole and walk straight south to the Equator, turn through 90° and walk east the same distance you have just walked south, and then turn through 90° and walk that distance once more, you will be back at the Pole, and have walked out a giant triangle—with internal angles adding up to 270°! There was surely something wrong with this state of affairs. You can’t just go ripping up ordinary geometry—aka Euclidean geometry, after the Greek who first discussed it—like that!
The attempted solution was to look for axioms— fundamental rules on which the more complicated rules, like the value for the internal angles of a triangle, are based. Sadly, the conclusion of this quest was abject failure. The final nail came when it was proved mathematically that finding a self-consistent, provable set of axioms is impossible. The conclusion was that there is nothing special about Euclidean geometry: in spite of its enduring popularity and elegance, there was no reason to choose it over any other self-consistent geometric system.
However, I seem to be stuck firmly in the early twentieth century: I am concerned that there is something a little bit special about Euclidean geometry. That something is that the properties of shapes do not vary depending on how big you draw them—a quality called ‘scale invariance’—whereas in other systems of geometry, they do.
So what on Earth does that mean? Well, imagine you’re on Earth, stood at the North Pole, and you want to draw a circle. You tie a rope around the pole (there is an actual pole at the Pole, right?), tie a pencil to the other end, walk far enough away that the rope goes taut, and then turn through 90°, put your pencil to the floor, and draw your giant circle.
For a normal circle, we know the circumference, c = 2πr, where r is the circle’s radius and π = 3.14159265358979…, which mathematicians have worked out to more decimal places than I care to reproduce here. What about on the surface of a sphere?
Imagine your string was long enough that you could walk to the equator—a quarter of the circumference C = 2πR of the Earth. The circumference of your circle is then C—the circumference of the Earth—because you’ll have to walk around the whole Earth to finish your circle. Let’s try to apply our favourite formula: I’ll call it c = 2π′r, this time;π′ because I’m worried the answer won’t just be π any more.
- C = 2π′C/4
- C = π′C/2
- π′ = 2
Damn! The value of π′ seems to have changed. Even worse, it will change for every different size of circle you try to draw. Imagine drawing a circle with a radius C/2—half-way around the Earth—you’ll walk all the way to the South Pole, and your circle’s circumference will be zero!
Luckily, having broken geometry, trigonometry can come to the rescue. We can actually work out how π′ changes, and then plot it out on a graph:
π′ gradually falls to zero—the case where your circle is a point on the South Pole; and then goes negative as you go around again (because you’re kind-of drawing the circle backwards…it is worth spending a moment thinking about this); and then comes positive again as you pass the North Pole again when walking out your radius. When you’ve walked 1¼ times around the Earth, your circumference is the same as when we worked out π′ = 2 before, but this time your radius is much longer, 5C/4, and so π′ has fallen to a mere 2/5. It gets smaller with every additional walk around the Earth: the radius r gets bigger and bigger, but the circumference can never be longer than C.
It really is worth grabbing a piece of paper and drawing yourself some pictures at this point to convince yourself that all this is correct.
image adapted from Wikipedia
The reason it’s worth making sure is because, just as we’ve digested this problem on the surface of a sphere, though, another type of geometry appears—hyperbolic geometry is geometry conducted on a surface which looks like a horse’s saddle. This time, instead of getting smaller, the circumference of a circle gets larger because you’re walking around the perimeter and up and down on the undulations of the saddle.
Indeed, if you draw a bigger circle,the circumference gets larger disproportionately quickly:
This madness doesn’t happen in Euclidean geometry—it doesn’t matter how big you draw the circle, π = π = π = 3.14159265358979…. This is that special scale invariance I am worried about.
Scale invariance may be an arbitrary quality to venerate in a geometry, but nonetheless Euclid’s is the only one to possess it. Perhaps bugs living on the surface of a sphere would think we were mad to consider it special, and instead other properties of shapes would be more significant for them. However, π would be a special number even for the sphere-bugs, for they would know that as they drew circles smaller and smaller, the ratio of the circumference to their diameter got ever-closer to 3.14159265358979…. The smaller the part of a sphere you look at, the more it looks flat, and the better the flat-plane geometry approximation gets.
It is for these two reasons that Euclidean geometry is a special case. There are many different radii of sphere or curvatures of saddle—but only one way to be flat. It feels like it should be accorded some philosophical significance, as though it is somehow deeper, or truer than spherical or hyperbolic geometries.
So should we reserve some reverence for it? Perhaps; or perhaps it is no more special than a square is a special rectangle with all the sides the same length, and there are (probably) no deep truths about reality hidden in squares. But Euclidean geometry seems somehow more elegant, more fundamental, than squares. I find its special-ness a little disconcerting, as though there is something there. A good many eminent philosophers and mathematicians about ninety years ago were with me. Are you?
Addendum for geeks
Maths nerds will enjoy working through the trigonometry and establishing that
π′ = π sin (r/R) r/R
for the case of a circle drawn on the surface of a sphere.
Even better, maths nerds will definitely appreciate re-writing these equations with the aid of a clever trick. You can define the curvature of a sphere as K = 1/R2. A really small sphere is really curved, so has a large value of K, whilst a larger sphere looks ever-more like a flat surface, and so is less curvy, and K gets smaller.
It turns out that a hyperbolic surface is mathematically identical to a spherical one with negative curvature, so K < 0. This has a very neat consequence. We can replace all the terms in the equation for the surface of the sphere with r/R with r√K. The general equation, then, is
π′ = π sin (r√K) r√K
This means that, if K is negative, then we get an imaginary value for √K, and thus the equation is transformed into
π′ = π sinh (r/R) r/R ,
where R = ik is the imaginary value of the ‘radius’ appropriate for a hyperbolic surface. This is the equation of the second graph above. Which, given that sinh is the hyperbolic sine, should perhaps not come as a surprise.
The general formula for π′(r, K) shows us that there are two limiting cases where π′ = π; π′(r, K = 0) and π′(r = 0, K). These are the limits of zero curvature (flat, Euclidean space) and zero radius (an infinitely small circle), respectively.