Why do 4 planes make 15 regions, and not 16?

Answer 1

Yes, this actually fairly easy to see if you do the following:
(I'll get to the 3D case shortly)
Draw two intersecting lines on a piece of paper(they don't have to be perpendicular). Call the point of intersection "p1". Now you could draw a third distinct line passing through p1 but the plane would only be divided into 6 regions. However if you draw the 3rd line so it intersects the first two lines at points p2 and p3, then the plane will be divided into 7 regions. The 7th region is the the triangular area enclosed by p1, p2, p3, and their connecting line segments. It is easy to see that this is the best you can do with 3 lines in 2-space. You will need 3-space to get 8 = 2^3 regions. So in the planar case the maximum is 2^3 -1 =7 Now imagine those 3 lines on the paper are planes that pass through the sheet of paper. Then those 7 planar regions will become 14 3-space regions (7 above the plane of the paper, and 7 below). And of course we now have 4-planes. But is this the best we can do with 4 planes? No. The easiest way to see this is to imagine 3 mutually perpendicular planes which create 8 regions. Now if we introduce the 4th plane so that it does not pass through the point of intersection of the 3 plans but it is perpendicular to two of them then we will only get 14 regions, as dicussed. But we can angle this plane so that it is perpendicular to none of the 3 planes and it cuts through two of the perpendicular planes (and not through the xyz origin) in such a way that we create two triangular wedge shaped regions (one in one of the eight regions and another in one of the other kiity-corner regions). This gives us one more region (tolal of 15). But again, as in the 2D case, it is clear this is the best we can do. We need 4-space to create 16 regions.
So the maximum = 2^4 -1 =15

Answer 2

I believe that 16 regions are possible because with 4 planes in the same space, I can have four regions (if they are all parallel) or eight regions (if they all intercept at a common line) or 12 regions (if they all intercept at only one point but any two share a common line like the XYZ planes we are familiar with). I am sure there are other numbers possible too if you just sit and play around with them and use your imagination.

Answer 3

Beats me. Do you mean four perpendicular planes in four-space which all share a point? If so, I would have guessed 16 as well. Are you sure it's 15? Can you give us more details?