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=26 3D: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.
Hint: Don’t read on below the line if you want to figure it out yourself. You are making your life more difficult by excluding the point where all coordinates are zero. If you include them, you get $3$, $9$, and $27$ points in $1$, $2$, and $3$ dimensions, respectively, which follows a clearer pattern.
You seem to be looking for the number of points where each coordinate is $1$, $-1$, or $0$. This is three choices per coordinate, so there are $3^d$ points in $d$ dimensions.
However, you’re excluding the all-zero point, so your numbers are one lower than that.
And finally, in zero dimensions, you decided to not exclude the zero point, so in that case, your number is not one lower after all.