Concrete mathematics : a foundation for computer science

(A pizza with Swiss
cheese?)
A region is convex
if it includes all
line segments between
any two of its
points. (That’s not
what my dictionary
says, but it’s what
mathematicians
believe.)
1.2 LINES IN THE PLANE 5
defined by n lines in the plane? This problem was first solved in 1826, by the
Swiss mathematician Jacob Steiner [278].
Again we start by looking at small cases, remembering to begin with the
smallest of all. The plane with no lines has one region; with one line it has
two regions; and with two lines it has four regions:
(Each line extends infinitely in both directions.)
Sure, we think, L, = 2”; of course! Adding a new line simply doubles
the number of regions. Unfortunately this is wrong. We could achieve the
doubling if the nth line would split each old region in two; certainly it can
split an old region in at most two pieces, since each old region is convex. (A
straight line can split a convex region into at most two new regions, which
will also be convex.) But when we add the third line-the thick one in the
diagram below- we soon find that it can split at most three of the old regions,
no matter how we’ve placed the first two lines:
Thus L3 = 4 + 3 = 7 is the best we can do.
And after some thought we realize the appropriate generalization. The
nth line (for n > 0) increases the number of regions by k if and only if it
splits k of the old regions, and it splits k old regions if and only if it hits the
previous lines in k- 1 different places. Two lines can intersect in at most one
point. Therefore the new line can intersect the n- 1 old lines in at most n- 1
different points, and we must have k 6 n. We have established the upper
bound
L 6 L-1 +n,
for n > 0.
Furthermore it’s easy to show by induction that we can achieve equality in
this formula. We simply place the nth line in such a way that it’s not parallel
to any of the others (hence it intersects them all), and such that it doesn’t go