Well it just seems a bit of a strange claim because, apart from various fields of theoretical physics which can
only be described using maths, people tend to try to understand the basic mechanisms and a kind of 'visual understanding' of something way before they try to model it and explain it mathematically. I don't understand the maths behind a hell of a lot of this stuff, but I'm fairly confident with a lot of my overall comprehension of the ideas.
You ask about polar patterns. Very basically (and typed quickly with lots of mistakes because I'm damn hungry and cooking dinner

)...
Firstly, remember that sound travels in the form of longitudinal pressure waves.
Omni mics are pressure transducers - the diaphragm is sealed on one side (apart from a small vent to allow for slow changes in atmospheric pressure) - and so the diaphragm responds equally to changes in pressure from all directions.
Figure-8 microphones are pressure
gradient transducers - the diaphragm is open on both sides and so responds to the difference in pressure on either side.
Cardioid patterns combine aspects of both of these. The diaphragm is 'partially' open at the back... vents / 'acoustic labyrinths' are used to mechanically delay the pressure waves reaching the back of the microphone, thus introducing a phase shift. Compressions or rarefactions reaching both sides of the diaphragm simultaneously will result in zero net force (e.g. sounds arriving out of phase will result in destructive interference).
For sounds arriving from the rear of the mic, the delay introduced at the back of the diaphragm will result in it arriving at the same time as the sound reaching the front of the microphone, but in antiphase, hence the 'rejection'. For sound arriving from the front the effect is more or less negligible, though this is related to how the proximity effect arises.
I did have some basic diagrams I sketched out for a physics assignment a while ago, but I don't know where they're saved on my computer at the moment.