What is the maximum number of regions in 3D space a plane can intersect?

Question

Since there are 8 regions or"quadrants" I thought it would be 6 regions as the max. I do not know if I am right.

Answer 1

Note, I am using the open definition of an octant, where the coordinate planes are not in an octant. If you allow the coordinate planes to be in an octant, points can be in multiple octants simultaneously, which is probably not what you want. Also, such a definition would allow a plane to be in 8 octants, if the plane is a coordinate plane.

To see that a plane can intersect 7 octants, consider the plane $x+y+z=4$. Each of these points are on the plane, representing 7 different octants:

$(2,1,1)$

$(-1,3,2)$

$(3,2,-1)$

$(2,-1,3)$

$(-2,-1,7)$

$(7,-2,-1)$

$(-2,7,-1)$

The 8th octant is where all coordinates are negative, which is clearly not possible with this plane.

Answer 2

If we exclude a plane through a coordinate axis, the equation is $$ax+by+cz+d=0,$$ where WLOG $a,b,c,d>0$. Then this plane cannot intersect the first octant, but it does intersect all others, because the negative coordinates can compensate the positive terms.