Geometry

External martial arts (because I don't know squat about internal) are all about geometry. No matter if it's karate (Goju-ryû, or Kyokushinkai), or savate, or... Deflection is geometry, footwork is geometry, even punching.

Which is why a lot of discussions on martial arts are plain dumb. Part of it is in the explanation. Martial arts are full of sentences that make no sense. "Do that without any strength, at all" At all? How can I stand upright, then? Yes, yes, obvious... until the example is not quite as clear. "Do X in a straight line" But... er... it's an arch.

And then it starts. And you get people who don't SEE anything beyond the name. People who don't see arches or circles, who watch you warily when you mention those, as if you were trying to mislead them.

Now, granted, some geometries are very small, subtle, difficult to perceive. Others are pretty large. And they could be taught. but teaching geometry goes against the grain of martial artists. Unless they go to structured extremes, where shapes are just this wide and just that size. Which is not, can't be, real: shapes will ultimately depend on the situation, opponent, your own body type and your mechanics. But the commonalities will be there. You'll have an arch, or a triangle. And the arch will be wider or shorter, the triangle will be squat or longish. But the mechanics of an arch don't deliver properly is you use a triangle instead. And counterwise.

And teaching that way would be more universal, less prone to sects and cliques, and...

Which is another reason it's not done, of course.

Take care.