Many theorems in mathematics say, in one way or another, that it
is very diﬃcult to arrange mathematical objects in such a way that
they do not exhibit some interesting structure. The objects in the
Erd˝ os distance problem are points, and the structure we are curious
about involves distances between points. We can loosely formulate the
main question of this book as follows: How many distinct distances
are determined by a finite set of points?
1. A sketch of our problem
In the case that there is only one point, we have but one distance,
zero. It might seem odd to count zero as a distance, but it will make
things easier later on if we just assume that it is. In the case of two
points, our job is pretty easy again. We have the distance between the
two points, and again, zero. However, if we consider the case of three
points in the plane, it begins to get interesting. Three points arranged
as the vertices of an equilateral triangle are the same distance from
one another, so there is only one nonzero distance, making two total.
If they are the vertices of an isosceles triangle, we have one distance
repeated, leaving three distinct distances total. Of course, there are
any number of ways for three points to determine four distances.
These phenomena increase in complexity and frequency as we consider
more and more points. In fact, there is no configuration of four points