Dr. N.K. sent me a little puzzle.
Construct the Fill of a set of points by drawing all the lines (extending to infinity) using every pair of points in the set. Thus the Fill of three non-collinear points is the triangle with the 3 points as vertices and the edges extended to infinity. The Fill * Fill of three non-collinear points is the whole plane.
The question is: what is the Fill * Fill of four non-coplanar points? Is it all of space (Euclidean 3D space)?
Answer beneath the fold.
The Fill * Fill of four non-coplanar points is not all of space, you have to exclude four points. If you take one point to be the origin and the three other points to define vectors a1, a2, a3. These three vectors form a basis, and any point can be expressed as
Construct the Fill of a set of points by drawing all the lines (extending to infinity) using every pair of points in the set. Thus the Fill of three non-collinear points is the triangle with the 3 points as vertices and the edges extended to infinity. The Fill * Fill of three non-collinear points is the whole plane.
The question is: what is the Fill * Fill of four non-coplanar points? Is it all of space (Euclidean 3D space)?
Answer beneath the fold.
The Fill * Fill of four non-coplanar points is not all of space, you have to exclude four points. If you take one point to be the origin and the three other points to define vectors a1, a2, a3. These three vectors form a basis, and any point can be expressed as
{x1,x2,x3} ≡ x1 a1 + x2 a2 + x3 a3
In this notation the four excluded points are:
{½, ½, ½ }
{-½, ½, ½}
{½,-½, ½ }
{½, ½, -½}
PS: more detail
