From Pizza Cut, we know the formula for maximum partitioning (pieces) of the circle, given n straight lines (cuts).
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1. Determine the maximum number of regions of the plane produced by n intersecting circles.

2. Determine the maximum number of regions of the plane produced by n intersecting ellipses.

3. Determine the maximum number of regions of space produced by n intersecting spheres.

The spheres are tricky cuz they are hard to visualize. I could visualize up to 4, but not 5. So I don’t know if my generalization will hold even to 5.

I don’t think that the answer is the same as for circles, though, which you can see after the fourth sphere is added. If you place your first three spheres such that the top view looks like a Venn Diagram (just like the circles), you can place the fourth sphere above the other three (such that it looks like a squashed pyramid). The fourth sphere can cut through all of the previous regions (the circle couldn’t do this – it would always miss one region, or miss 2 regions but add another in space) plus make a new region in space. I don’t know how to explain that better.

Spheres:
n      regions added      regions     
0                                    0
1               1                   1
2               2                   3
3               4                   7
4               8                 15

Based on this, it seems that you are adding 2^(n-1) each time.  So the formula is [sum of 2^(n-1)] = [(2^n – 1)/(2-1)] = 2^n – 1.

Like I said, I don’t know if it’s right.

Posted by nikki on 2004-09-14 10:51:42