Take any four points in space. Draw all lines connecting pairs of them. Then draw all lines connecting pairs of points on those lines.

Can the resulting set of points cover all of space?

It's clearly true for 3 points and the entire plane defined by those 3 points, if I understand the problem correctly.  Just picture a triangle with the sides extended out to infinity, then connecting points of one line with points of another line would "color in" all the area between.

I'm not sure I can make the jump to 3 dimensions.  Picture the original triangle of points, then one more point that is not coplanar.   Now there are 4 triangles, the faces of a tetrahedron.  It seems clear that all the points lying in each of those 4 planes can be "colored in".  But to fill in all space I can't see intuitively or prove.

Posted by Larry on 2005-02-06 16:43:33