I think the universe is pure geometry - basically, a beautiful shape twisting around and dancing over space-time.
- Antony Garrett Lisi
In his book, The Age of reason, American philosopher and founding father, Thomas Paine, argued about the authenticity of the Bible as the words of God. He went a step further by arguing that he has no way in proving the authenticity of The Republic as Plato's words either. However, there is one work that he is certain to be authentic and that is, Euclid's elements. In 13 books Euclid has written and proved the basis of geometry, the mathematics of shapes and forms. Euclid's elements have been studied for two millenia by the greatest of minds. Indeed, Newton's Principia is full of drawings and geometric interpretations of the world and mathematics.
As geometry proved to be more and more useful in studying nature, euclidean geometry was not enough. Euclidean geometry described only flat space and this caused quite a few problems. The surface of the Earth can be approximated to a sphere(a Two dimensional surface) but in order to make maps, the shapes and sizes of continents have to be deformed and you'll never have a complete map (Have you noticed that most maps don't show the whole of Antarctica?). There is a simple way to solve this problem, by embedding the surface into a higher dimension. Thus, instead of making maps, we can make globes (A three dimensional ball). Embedding manifolds into higher dimensions have limitations and it is not always elegant. In the 19th century, mathematicians Gauss, Bolyai, Lobachevsky and Riemann independently began developing new geometries, non-euclidean geometries. For example, using Riemannian geometry it is possible to describe the two dimensional surface of the Earth without introducing a third dimension. Non-euclidean geometry proved to be useful in Einstein's theory of general relativity. It was the comeback of geometry in physics.
Why geometry had to do a comeback? Well, Leibniz's formulation of calculus was considered to be more useful than Newton's. With Leibniz you could have coordinates (developed by René Descartes), equations, numbers to calculate... while with Newton, you had lines, circles, shapes... The development of topology (the study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures) and differential geometry (+differential topology) showed that geometry could be as useful (or even better) at understanding nature rather than analytics. The comeback of geometry can be attributed mostly to French mathematicians Henri Poincaré and Elie Cartan.
So far, geometry has shown its importance mostly in classical general relativity while in quantum mechanics, probabilities and time-dependent differential equations still rule. Can we rewrite quantum mechanics using geometry? That's the question I'm trying to answer. The problem I'm trying to solve. Whether it is string theory or quantum loop gravity or anything else that is going to lead to the theory of everything, it is clear that geometry has an intrinsic part to play in it.