- 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. ...
> 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.
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?
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.
C'est la vie.
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.
10 years ago PERC cells weren't available on an industrial scale, even though the technological basics were discovered, explored at the lab scale, and published in the 1980s. It took a lot of manufacturing advances and market evolution before PERC technology was both practical and profitable to manufacture for large scale use.
http://www.aleo-solar.com/perc-cell-technology-explained/
Likewise, I expect that some battery ideas that are published and "go nowhere" will eventually reach industrial scale, but only much later.
We have been spoiled by the web to expect a whole other time scale, but physical technology still takes the time it has to take.
My - not very informed - impression is that battery technology actually is moving very fast, considering the timescale constraints.
That's precisely the sort of thing that most manufacturers don't want, because a battery designed to last effectively forever means less recurring revenue on replacements.
"100% recyclable" is good for them (and "biodegradable" even better), because they can act "green" while continuously making products that don't last and have to be recycled, expending even more energy and creating profit in the process. "The best kind of planned obolescence is environmentally friendly planned obolescence."
That's ridiculous, tinfoil hat thinking. Longer cycle life = cheaper battery = higher profit + happier consumer.
Start by reading about the Phoebus cartel: https://spectrum.ieee.org/tech-history/dawn-of-electronics/t...
I remember my first electric razor. It failed after a couple of years, so I took it apart to find out why. I discovered that the electrical contacts to the motor were just little pieces of graphite, and when they wore down to nothing the razor was finished. Definitely planned obsolescence in action!
(Search YouTube for "vintage induction motor" and you'll find plenty of century-old(!) examples still in good running condition. I don't think the same can be said of the brushless motors today.)
https://en.wikipedia.org/wiki/Brushless_DC_electric_motor
AFAIK there is no substitute for graphite in a brushed motor, it is needing to transfer power to different sections of a rotating part in turn and does so by rubbing over a set of copper strips. That the graphite is soft is why it works well but also why it wears out rather than the commutator, which will survive several sets of brushes.
Those breakthroughs, when applied to scaled up battery manufacturing give us the 5%-10% compounding annual improvement we see. A doubling at least every 15 years is pretty good!