I am working on a presentation that shows the exponential increase as one increases the number of dimensions, and I'm trying to figure out a way to calculate all non-zero or null counts, which I'll call "points" here.
For ease of understanding in the presentation, I will only be using full count of one in any direction. So nothing beyond one and no fractional numbers. Here's what I have so far...
In zero dimensions, there's nothing to separate from anything else, so we start with everything, which is one.
0D:1P
In one dimension, you separate it into positive and negative along a line. So, there's two points separated by zero.
(X-1;X+1)1D:2P
In two dimensions, it gets interesting, because not only do you have positive and negative of the X and Y, but also the other points that are combinations of the two. So there's 8 points and a null center (X0:Y0).
(X-1:Y-1);(X-1:Y0);(X-1:Y+1);(X0:Y-1);(X0:Y+1);(X+1:Y-1);(X+1:Y0);(X+1:Y+1)2D:8P
In three dimensions it's still fairly easy, as the points can be measured with adding three two-dimensional sets together (for Z-1,Z0,& Z+1 sets) with an extra two for the non-null points (X0:Y0:Z-1 & X0:Y0:Z+1). Again, only the null point (X0:Y0:Z0) isn't counted. This I'm putting here as a simple equation to save space.
8×3+2=263D:26P
Now, my question is, how does a person figure out the number of points from there without counting each individual point?
Ideally, I'd like to find out up to 12D to have a good spread to possibly graph out, but even if it's some sort of equation that can show the number of points in Nth dimensions it'd be awesome. Thanks in advance.