- Al Bartlett
If we thought in 100D, we might have a better sense for it, because we'd be able to see a hundred of them.
Hypervolume grows exponentially.
One way to get a really rough idea is to try and control each and every joint individually.
Close your eyes and try to imagine that each joint, each muscle is a dimension along which you can move (by moving it), and your posture at any given moment is a point in that space. When you move, you make a line through it. Don't picture it, just feel it.
What is the shape of that space?
You can get an idea of what exponential growth is like by exploring how the shape of that space changes as you add more and more things you're controlling.
> ... In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. ...
I never completely figured out Aikido with it’s joint locks and levers. Maybe talented aikidokas have a grater capacity to visualize/fill this type of activity?
Interesting point, but I don’t think Aikidoka have any special talent for that: we use a small number of techniques and what changes is the way you use them in response to different attacks/holds.
Also, you tend to work on your specific Ryu (school) technicsl curriculum and nobody goes around “inventing” new locks.
(Some argue that Aikido is not really adapting to modern world nor cross-pollinating with other martial arts due to -arguably excessive - reverence for tradition).
You can model what you're doing as a phase space, which is the product space of the state of each thing you control. This generally has a lot more dimensions than three. (You see this in robotics; a 5-axis CNC has a 5 dimensional phase space for position (5 axes of motion), plus a few more dimensions for things like speed and coolant flow.)
That mashed up with the meditation idea of starting with your focus on something really small -- the soles of your feet, for instance -- and drawing it up your body until you can feel all of it.
If you do the two, you can slowly draw yourself into awareness of higher and higher dimensional phase spaces, which shows you a curve of exponential growth.
Well, okay -- I also followed Terry Tao's excellent advice on dealing with higher dimensions, to stop trying to picture math and start trying to find systems that expressed it in what they did. You can often get a feel for a system doing something more complex than what you can directly picture.
