planar divisions? urgently need answer!?

Question

at maximum how many regions in space can be created by the intersection of 5 planes? answer is 26, need to know why? prove it please!! 10 points for steps leading up to right answer! proof needed

Answer 1

Unfortunately, sketches are seldom helpful in 3-D geometry. The only way I can do this stuff is by mental images. Do this one plane at a time.

The first plane divides the region in two.

The second plane either intersects the first or it does not. If it does not intersect, it will split only one region in two for 3 regions. If it does intersect, it will split both regions in two giving four regions.

In general, each added plane must pass through as many pre-exisitng regions as possible to create a maximum number of regions for the next plane to intersect.

The 3rd plane intersects both earlier planes and splits all 4 regions for 8 regions. Topologically, the system of regions is equivalent to three orthogonal planes. The actual angles, etc. are unimportant. All three planes must intersect at a single point. Further, all 8 regions meet at this intersection. Call this intersection P1.

The 4th plane either passes through P1 or it does not. If it does, it will split no more than 6 regions. Any general plane (not parallel to any of the lines formed by intersections of previous planes) that passes through P1 will split 6 regions. The two regions that are not split are the two that contain the normal to the 4th plane at P1. By moving the 4th plane any arbitrary distance along one of these normals. the plane no longer passes through P1 but does split one more region for a total of 15.

In displacing the 4th plane, you have created 3 new points (P2, P3, P4) similar to P1 where three planes intersect. There are four of these triple intersection points. Each point has 8 regions meeting at it. The fifth plane acts much like the fourth plane relative to each intersection point in that it is not possible to split all 8 regions. One must be left unsplit for each intersection point so only 11 of the 15 can be split for 26.

Answer 2

The best way to prove this is with a drawing. Start by drawing two planes, but, don't make them perpendicular. Instead, make the angle as small as possible. Now, draw a third plane such that it will intersect the first two at different lines. You lose a region if it intersects them at a common line. Remember that your planes and lines will be represented by lines and points on your paper 2-dimensional model. Make the fourth plane intersect the prior three again without a common line between any two and the fifth intersect all of the prior four. This should maximize all of your regions.