Cutting up a hypersphere
I set myself some puzzles to solve, but I haven't been able to work out this one and was wondering if someone could point me in the right direction.
Basically, I wanted to see what the maximum number of areas I could divide a circle into with N number of cuts, then after that worked out the same thing but for cutting up a sphere with 2D planes. I managed to get these right and have confirmed it with other sources:
For cutting circles, see: Circle Division by Lines -- from Wolfram MathWorld
And for cutting spheres (or in this case cubes), see: Cube Division by Planes -- from Wolfram MathWorld
Lastly (and the one I'm currently stuck on), I want to work out the maximum number of areas N three-dimensional planes can divide a 4D hypersphere into. I have an answer, but it doesn't make much sense and I'm already out of my depth, so I was hoping someone could help me out.
The answer I've come up with is as follows:
The function G(n) gives the number of areas added by the nth cut: G(n) = 1/6[(n-1)(n-2)(n-3)+6]
So in order to see the maximum number of areas, say, 4 cuts would produce, you would need to add up:
G(4) + G(3) + G(2) + G(1) + G(0)
= 10 areas
The reason I suspect this is wrong is because (among other things...) it produces less pieces than 4 two-dimensional cuts would on a 3D sphere, but I would expect higher dimensions to produce more areas.
Any help much appreciated!