# What are the ways 3 planes in $ℝ^3$ intersect?

Will they be able to form points, lines, or planes?

In my opinion, the planes can only form points and lines, could someone give any counterexamples?

• Look at this image for some insight into the different ways that three distinct planes can intersect in $3$-space. Try to figure out what the intersection of each is geometrically.
– user137731
Dec 21, 2015 at 1:45

1) The three planes can be parallel. And not intersect at all.

If two planes intersect the intersection will be a line.

2) Two planes can be parallel and the third plane intersects each. The third intersects each at a line. These to lines are parallel and co-planer.

3) All planes intersect at a line and the third intersects the two on the same line (like pages in an open book intersecting at the spine).

4) The two planes intersect and a line. The third intersect each at a parallel angle to insect at second and third parallel line. The planes will from a triangular cylinder, with each pair of planes intersecting at a line. These three lines are mutually parallel and non planar.

5) General case. Two planes intersect. A third intersects obliquely and the three intersect at a point. Each pair of planes intersect at a line. The three lines are neither coplanar nor parallel and the three lines intersect at the point where the three planes do.

Two distinct planes intersect in a line or not at all (prove this!). The line can intersect a third plane in either a line or a single point (prove this!). It's easy to construct examples where each of these cases can happen.

Now if two or more of the planes coincide, there's one more case.

• Or the third plane might not intersect the two two itersecting planes at all and eah pair intersect mutually but not together forming three parallel non planar lines. Or two planes may be parallel and the third intersect both. Or three planes may be parallel Dec 21, 2015 at 1:37
• So is it possible that the three overlap and form a plane? Dec 21, 2015 at 1:39
• I'd answer that no. If the overlap, then they are both the same plane. Two distinct planes are either parallel and don't intersect at all or intersect at a line. Three planes can mutually intersect but not have all three intersect. The planes will then form a triangular "tube" and pairwise will intersect at three lines. This lines are parallel but don't all a same plane. Or three planes can, like the pages in the spine of a book, can intersect in one single line. I'll try to formalize this later. Dec 21, 2015 at 1:51