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?

Nevermind! There is little here worth reading.  See the posts of Tristan and Charlie for the real story.

Let the 4 points be the origin O and the unit points X,Y,Z of the coordinate axes. Write OZ as the set of all points of the form (0,0,z),  and write XY as the set of all points of the form (x,1-x,0). Then the join of OZ and XY will be the set of all points of the form (ax,a(1-x),(1-a)z). If we then require ax=t, a(1-x)=u, (1-a)z=v, we get a=t+u, x=t/a, z=v/(1-a) so long as a is neither 0 nor 1 (i.e. t is neither u nor 1-u). Hence if a is not 0 or 1,  there then is a corresponding point on the join of OZ and XY. But if a=0 or 1, an infinite coordinate will usually result.

The choices of points used above are sufficiently general to show that some points cannot be generated in the manner of the problem if only two lines are used as generators. One more step of joining would finally prevail, however.  We need to join with lines all the points that we get in the manner of the problem and then all these resulting points will finally cover space.  With only two stages of joining, we need the points at infinity on the lines.

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Edited from earlier versions which incorrectly claimed that all points of space could be generated from the two lines. The present version is mistitled and is no solution at all since only two generating lines are considered, not six.

Edited on February 7, 2005, 12:47 am

Posted by Richard on 2005-02-06 18:41:27