MANIFOLDS

I would really like draw a picture of the fabric of space, but the math needed to model space-time is too complicated.  We can think in terms of the three space dimensions, but when we start to go into higher dimensions the models just do not translate in our brains.

Enter the mathematicians.

Euclid of Alexandria wrote "Elements" somewhere around 300BC.  The fact that you can find Euclid's work repeated in any text book of geometry and algebra attests to it's remarkable impact on the sciences.  In some ways, you can also see a beauty to the progress of Euclid's mathematical formulations.  Euclid probably took many of the concepts and proofs from other mathematicians of the time, but he was the first to put together all of that knowledge into writings that survive.  Euclid started with five main postulates upon which he built his mathematical model of the world.

1. Any two points determine a unique line containing them.
2. Any line segment may be extended.
3. Given a point P and a distance r, there is a circle with center P and radius r.
4. All right angles are equal. 
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Euclidean Geometry was the only game in town for a couple thousand years, but that was not for want of trying. Many mathematicians took a shot at postulate number five, which did not follow directly from the previous four. Euclid saved the fifth postulate till last for a reason. He obviously had problems with it himself, and by definition, the postulate has some esoteric flaws. Take perspective; when we view a railroad track disappearing into the distance, we understand that the tracks are parallel - but that they appear to converge to a point. Can we model visual perspective as a geometry of it's own? The answer is yes, but for centuries Euclid's geometry held it's own. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry, primarily through the works of Johann Carl Friedrich Gauss (who never published his theory), János Bolyai, and Nikolai Ivanovich Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for the development of geometry, and Jules Henri Poincaré, who was looking for the math to back up his theories of relativity. 

Hyperbolic space is a geometrical space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Mathematicians would have been content to just play around with the math, but along came a German mathematician, Georg Friedrich Bernhard Riemann, whom was not content with the status quo. Riemann came up with a new geometry using “tensors”. He also found a way to relate his new geometry to four dimensional analysis.  His broad vision of the subject of geometry was expressed in his 1867 inauguration lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, a mainstay of modern geometric analysis.

Then in 1896 Pieter Zeeman and Hendrik Antoon Lorentz used post Euclidean geometry to accurately model a part of the natural world, magnetic spectral lines, and the door was open. Lorentz went on to work with a young physicist, Albert Einstien, on a new theory of relativity. By 1906 it was noted by Poincaré that, by using an imaginary time coordinate √−1 ct, the Lorentz transformation can be regarded as a rotation in a four-dimensional Euclidean space with imaginary time being the fourth dimension, "manifold space".

The concept of manifolds is important because you can model complicated structures in terms of the relatively well-understood properties of simpler spaces. For example, a manifold is typically endowed with a differentiable structure that allows one to do calculus and a Riemannian metric that allows one to measure distances and angles. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism (Sir William Rowan Hamilton) of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity.

Using the Lorentz transformation, you can mathematically model how two observers' varying measurements of space and time can be converted into each other's frames of reference.

This idea was elaborated by Hermann Minkowski who used it to restate the Maxwell equations in four dimensions showing directly their invariance under Lorentz transformation. He further reformulated in four dimensions the then-recent theory of special relativity of Einstein. From this he concluded that time and space should be treated equally and so arose his concept of events taking place in a unified four-dimensional space-time continuum. In a further development he gave an alternative formulation of this idea which did not use the imaginary time coordinate but represented the four variables (x, y, z, t) of space and time in coordinate form in a four dimensional affine space. Points in this space were regarded as events in space-time. The spacetime interval between two events in Minkowski Space is either space-like, light-like ('null') or time-like, creating Nutches for Nitches to fill.

Orbifolds

Phase space