schoolgirl problem (see page 167). In the theory of designing experiments the
Fano plane appears as a protean example, a Steiner Triple System (STS). Given a
finite number of n objects an STS is a way of dividing them into blocks of three
so that every pair taken from the n objects is in exactly one block. Given the
seven objects A, B, C, D, E, F and G the blocks in the STS correspond to the lines
of the Fano plane.
Pascal’s theoremA pair of theorems
Pascal’s theorem and Brianchon’s theorem lie on the boundary between
continuous and discrete geometry. They are different but related to each other.
Pascal’s theorem was discovered by Blaise Pascal in 1639 when he was only 16
years old. Let’s take a stretched out circle called an ellipse (see page 89) and
mark six points along it that we’ll call A 1 , B 1 and C 1 and A 2 , B 2 and C 2. We’ll call P
the point where the line A 1 B 2 intersects A 2 B 1 ; Q the point where the line A 1 C 2
intersects A 2 C 1 ; and R the point where the line B 1 C 2 intersects B 2 C 1. The theorem
states that the points P, Q and R all lie on a single straight line.
Pascal’s theorem is true whatever the positions of the different points around
the ellipse. In fact, we could substitute a different conic in place of the ellipse,
such as hyperbola, circle, parabola or even a pair of straight lines (in which case
the configuration is referred to as ‘cat’s cradle’) and the theorem would still be