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.
Neat, huh?
I remember discussing this with someone a long time ago and for some reason my way of thinking all this through doesn’t seem to go down well with mathematicians. Euclidian geometry seems to work perfectly (triangles with 180-degree internal angles, pi always being pi) as long as you stick to 2 dimensions. In 3 dimensions it’s just wrong (unless you have an infinitely small circle, blah, blah), often a good approximation, but nothing special. The universe has 3-dimensions that people easily cope with and therefore how can Euclidian geometry be considered a special case, since it is never right? Granted nearly two-dimensional systems can appear, but to be truly 2D is thermodynamically unstable and therefore Euclidan geometry is always wrong. Comparing Euclidian (a 2D general geometry) to spherical or hyperbolic (both 3D and for specific cases) doesn’t really make sense.
Maybe I don’t understand the argument completely but why should it be treated as a special case, in 3D? Also why did argument annoy my mathematician friend? That has remained more of a mystery to me than variable values of pi.
Some nice maths by the way.
You can do similar stuff in three dimensions, where you compute the surface area of a sphere and thus effectively re-define our favourite variable constant with A = 4π′r2, where A is the surface area of the sphere over which you make the measurement of 3D Euclidicity.
Obviously the hyperspatial diagrams in this case are slightly trickier to draw.
No idea why the mathematicians object…any mathematicians reading this who might know?!
The physical impossibility of realising two dimensions is also interesting. I am not quite sure what it means philosophically, though. Reconciling the somehow idealised forms of geometry with physical reality is an age-old philosophical difficulty: can the perfect shapes of geometry be said to exist in any sense separate from reality, or does the nature of reality inform the geometry we create? How the fact that modern physics has revealed that true two-dimensionality is an unattainable ideal fits into that…er…dunno.
The pragmatic scientist in me thinks you’re probably right, and that what we’ve got here is yet another limiting case.
Nope, afraid I don’t agree about Euclidean geometry being special. All your examples; excess of the sum of angles in a triangle and the excess of the circumference of the circle are all basically proportional to the Riemann tensor. Also your examples are just constant curvature spaces so the Riemann tensor is basically characterised by a single number; so really what you are talking about maps directly to real numbers. In which case your article amounts to saying zero is a special number. Which I don’t think we really can give much physical significance to.
You can try arguing that flat space is somehow special since all manifolds are locally flat. However, this is just basically due to the usual fact that if we do some small change in something we will get a linear change. This is exactly like approximating a curve, at a particular point, by a line.
I think really if you want to see something weird, along these lines, you should check out exotic R^4. In fact there are an infinite number of differential structures you can put on R^4, which if you think about doing a path integral over space-time manifolds you’re going to get a big contribution from four dimensional manifolds. Which is interesting.
I confess to knowing bugger all about the Riemann tensor (can an appropriate Riemann tensor with elements ∈ ℂ represent space with an arbitrarily complex curvature?), but: have you not just re-stated the case in a more mathematically rigorous, elegant, general and fundamental way?
Zero is a special number, and so though perhaps that fact is the most distilled formulation why Euclidean space is special, it is special nonetheless. I would perhaps go out on a limb and say we should worry wherever zeros appear in maths or physics…perhaps also ones, and definitely infinities.
ℝ4 sounds like a laugh—though I bet, even if I do get my head around it, it will be harder to turn into a popular science introduction than this was.
the argument above is completely wrong as u can imagine any circle drawn on the sphere, like yours which so happens to be the arc of a great circle, lying in a plane, so this argument states that pi=3.14=2=piprime!?!?
Its a blatent contradiction, so piprime =3.14 as well.