It's because we think in 3D, so we only really see three steps of exponential growth.
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. ...
That is growing exponentially with number of dimensions. The earlier post assumed fixed but large number of dimensions and varying linear size, so that would just be a power law.
> It's because we think in 3D, so we only really see three steps of exponential growth.
> If we thought in 100D, we might have a better sense for it, because we'd be able to see a hundred of them.
This is pretty clearly talking about getting a better sense of the asymptotic behavior in number of dimensions, and having a better intuition if you see a hundred steps than if you see three. The three steps of exponential growth mentioned are in transitioning from a single cube, to a line of 10 cubes, to a grid of 10x10 cubes, to a block of 10x10x10 cubes. But that's sort of where we tap out, because we're so heavily wired for 3D -- if we dealt with 100D, we'd have 100 such steps we could intuitively observe, and so have a better sense of asymptotics.
This is further seen in that the exercise is based on increasing the number of dimensions to explore the growth of the space as the dimensionality changes. It's literally adding more and more terms to a product space, and so clearly dealing with issues about dimensionality.
You're simply wrong, and incredibly uncharitable in your interpretation.
Further, geometric growth isn't a power law -- it's exponential growth. So the person asking the question was indeed confused, regardless of the fact you're wrong about what I was talking about. Geometric series are r^1, ^2, r^3, etc while a power law will look like 1^x, 2^x, 3^x, etc. Asking if an exponential growth is "just geometric growth" is being confused -- they're the same thing.
Oh oops, you’re totally correct that geometric growth is a synonym for exponential growth. I don’t know what I was thinking, but I may have been confused by the fact that volume grows polynomially faster than surface area for any fixed n dimensions (I now fear I may also be incorrect about this claim, although I’m confident it’s true for n=3).
Still, geometric growth is exponential in n when n is the number of dimensions, which isn’t really the n we were talking about in this context.
I discuss how seeing 100 steps of a sequence with regular behavior gives you a better sense of its asymptotics than seeing 3 steps, and then how you can generate some steps of that sequence as a mental model.
The N that is changing is the number of dimensions, both in comparing which model gives better asymptotic intuition and in terms of constructing a phase space by adding a dimension at a time.
I'm actually unsure how you could think there's an N that's not dimension, given that the only values discussed (or changing) were dimensions.
Did I not use fancy enough language when making a point to laymen, so you assumed you knew more than me and took a really uncharitable read so you could "correct" me?
When you say “volume grows” most people assume an increase in 3D volume. If you meant something else, like growth with number of dimensions, then you should’ve clarified it better. I think that’s why you are being downvoted despite the insightfulness of your comment.
It's actual a real problem with people who are moderately good at math:
They sabotage explanations to laypeople by incorrectly nitpicking technical details because they hear informal language that sounds similar to something they know, and rush to regurgitate that fact as a "correction" without really understanding the conversation -- and will insist on doing so unless you use language too sophisticated for the audience you were trying to reach in the first place.
This actually happens with nearly every field, I just experience it most with math -- it's probably related to Dunning Kreuger or whatever.
An interesting way to look at human thinking patterns. Is there any book on it?
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?
(Aikido SanDan, ~28 years of practice, still going to the dojo 3 times a week).
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).
Honestly, it's just a mash-up of a few Buddhist ideas and a few math ideas:
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.
