Apparently the “extra” dimensions of string theory are only supposed to be a few millimetres thick.

If that’s the case, could you dodge a bullet by moving a millimetre in the 10th dimension?

I guess it would depend on how wide the bullet and your liver are in the 10th dimension. Could lead to an interesting superpower: move in hidden dimensions.

The hero wouldn’t be invulnerable but would be less vulnerable. Still get an exit wound but maybe she’d only be grazed through the interior rather than completely ripped to shreds.

Still worth dodging/blocking a fist in the normal-sized 3 dimensions, but even a “direct” uppercut or body blow could become more of a “glancing blow” if she dodged in the thin directions. (NB: If ∃ 7 extra thin dimensions, each 1mm wide, and she dodged at once to “the other side” of all 7 at once—assuming, as well, that we’re “all the way to one side” of each of the extra dimensions—then she’d have made a total distance of √7mm between her and us.)

Joint locks—could she put someone in a joint lock they couldn’t get out of? Couldn’t she also get out of joint locks that no-one else could?

Couldn’t become invisible but become less visible.

Couldn’t pass through walls but could reach into crevices easier.

Could swim faster (twist her torso in the 10th dimension so the hands & feet still pull water, but less resistance on the mass of the body).

I used to think vectors were the same as points. They’re lists of numbers, right? Tuples with each item ∈ the same domain (or same data type). But … they’re not.

The way vector mathematics describes the forces of the world is to instantiate one linear-algebra at each point on the surface where a force is imparted.

When I strike a ball with my foot, look at the exact contact point (and pretend the ball doesn’t squish across my foot—even though it does, because my pegas are so poderosas). There’s a tangent space at that contact point and the exact particulars of my posture, shape of my foot|shoe, and so on determine the force vector that I use to bend the ball like Beckham.

Vacuum isn’t possible: Vacuum doesn’t occur, but hypothetically, terrestrial motion in a vacuum would be indefinitely fast.

So we know we only need that one strike and then inertia minus drag convolved with lay-of-the-land will determine a full path for the ball: till my teammate’s head contacts it—another tangent space, another vector—and we score.

That’s football for simple tangent spaces—the Aristotle-v-Newton scenario. Now onto the most poetic sport, where two holonomic dynamical systems dance within each other’s gyre, moving in and out of each other’s movements and—occasionally—osculating a tangent that links one part of me with one part of you.

To talk about striking different points requires a connection (parallel transport). [And remember if you hook my cheek versus jab my forehead, those arenotparallel so even more maths is involved. Also hard tissue (bone) versus soft tissue (cheek).]

Striking the ball on the side (to spin it) or under (to chip it) would be a connection on the S² manifold of the ball — moving the point of tangency — which is a different algebra’s worth of logic. Landing a punch on a different part of me is also a connection (that would be parallel transport of the strike vector on the surface manifold (with boundary) of my face/torso/armpit/whatever).

Keeping my torso over the ball is even yet another calculus—one that I suspect is much, much more complicated. Even though Julian James Faraway and others work on these questions of whole-body mechanics, I don’t think ∃ a complete mathematical theory of how all of the fixed holonomic parts of bodies that have very similar morphogenetic shapes (similar ratios of forearm to humerus length, etc) interact—how soft tissue and hard tissue in the usual places interact and specific angles can make a judoist with excellent technique and little strength able to whirl the body of a weight-lifter around her fulcrum.

Or how this guy can deliver a lot of force with proper dynamical posture (“shape”) when he’s clearly weak and fat. I can start to imagine the beginnings of something like that but it doesn’t obviously fit into the tangent space points & vectors story, except in a very complicated way of vectors connected to vectors connected to vectors, with each connection (not the same as the parallel-transport connection sense of the word I used above! Back to the base namespace) holonomically constrained or even "soft-constrained" in the case of soft tissue. Same with a blow landing parallel-transported different places on the surface of my body. The transport takes care of the initial strike vector but not how those forces travel and twist through my skeleton, soft tissues, down to the floor (either through my feet or through my *rse if the blow knocks me down). And sure, you could FEM-approximate my interior with a 3-D mesh and a deep geometric logic—and even make the ribs more rigid than the liver—but that’s just going to make you want higher maths even more as you grapple with the army of ε’s you’ll convoke with all the approximations, dx size, parameters, and so on.

Or why it’s better to swing the bat this way, or teach your body to do its swimming strokes in that motion, or various other dynamical or static particulars of human body shape and internal anatomical facts.

BTW, an amateur ultimate fighter once told me that the order of importance for winning a fight is:

technique (brasilian jiu jitsu technique)

balance

agility

strength

He didn’t mention tolerance for getting punched in the head but he seemed to put up with it remarkably. PS: judo guys have sexy bodies.