We are all taught the Theorem of Pythagoras at school and unless destined to become a physicist, architect or builder probably never want to hear about it again. We might remember that it says that: In a right-angle triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The ancient Greeks could prove it in various ways but agonized somewhat because they could not find the hypotenuse to be a fraction of whole numbers if the other two sides were equal.
This theorem, whether discovered by Pythagoras or not, is vastly more important than being just about triangles. It is probably the most important concept that we know because it underpins the whole of the structure of the physical universe, not just in theory but in fact. Of course, the universe is not made of triangles, right-handed or otherwise. So what is Pythagoras Theorem really?
The version we are taught at school is just the very simplest version which is only true in two dimensions on a flat surface. It retains much the same form if extended to real space which is curved and has at least four dimensions. It just looks a lot more complicated that’s all. Einstein’s theories of Relativity; Special (quite complicated) and General (very complicated) are really just fancy forms of Pythagoras Theorem. Hence this theorem not only underpins relativity but also its consequences, gravity for one. That you don’t fall through the ground to be fried at the centre of the Earth is another. The reason for the latter is a force called Electron Degeneracy Pressure. To understand this needs some knowledge of quantum theory and special relativity but is fairly easy to put into words. A future article maybe! Back to Pythagoras, or at least his theorem.
Then what is Pythagoras Theorem really? It is what is called a Metric. A Metric is a thing which provides a notion of the distance between two points in real space or abstract space. The hypotenuse is the only important part because it defines the position of the two locations that we want to know the separation of. The other two sides simply provide an arbitrary set of co-ordinates and there are an infinite number of different possible right-angle triangles of any particular hypotenuse. If space is curved we need to measure the distance between two points by adding together the longest sides of very many, very small (almost flat) triangles. That is why the General Theory of Relativity is so difficult.
But with the usual theory of relativity something rather remarkable happens. There are four squared terms in that version of Pythagoras Theorem since spacetime has four dimensions and three of them are preceded by minus signs. This means that under certain circumstances they can cancel out and so when travelling at the speed of light, the time measured by a clock on a spaceship travelling to a distant galaxy is zero! And a heartbeat is a clock. No need to go faster than light to explore the universe but the folks back on Earth will never find out about it. More details in my other article on Relativity.