214 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures
FIGURE 14.4. The Tictactoe plane.they showed that this cannot be done. Well, this is true in our world, and
since our world is finite, it is easy to check that it is true.” (We hope our
readers will accept the challenge.)
“We have 3 points on each line just as in the Fano plane, but we have
4 lines through each point—more than those Fanoans. All of our points
are alike, and so are all of our lines (even though the way you draw them
in your own geometry seems to differentiate between straight and curved
lines).”
FIGURE 14.5. Extending the Tictactoe plane by 1 line and 4 points at infinity.When we point out how the Fanoans love their property (b), our Tictac
guide replies, “We could easily achieve this ourselves. All we’d have to do
is add 4 new points to our plane. Each line should go through exactly one
of these new points; parallel lines should go through the same new point,
nonparallel lines through different new points. And we could even things
out by declaring that the 4 new points also form a line. We could call the
new points ‘points at infinity’ and the new line, the ‘line at infinity’ (Figure