28 Discrete geometry
Geometry is one of the oldest crafts – it literally means earth [geo] measuring [metry].
In ordinary geometry there are continuous lines and solid shapes to investigate, both of
which can be thought of as being composed of points ‘next to’ each other. Discrete
mathematics deals with whole numbers as opposed to the continuous real numbers.
Discrete geometry can involve a finite number of points and lines or lattices of points –
the continuous is replaced by the isolated.
A lattice or grid is typically the set of points whose coordinates are whole
numbers. This geometry poses interesting problems and has applications in such
disparate areas as coding theory and the design of scientific experiments.
The lattice points of the x/y axes
Let’s look at a lighthouse throwing out a beam of light. Imagine the light ray
starts at the origin O and sweeps between the horizontal and the vertical. We can
ask which rays hit which lattice points (which might be boats tied up in the
harbour in a rather uniform arrangement).
The equation of the ray through the origin is y = mx. This is the equation of a
straight line passing through the origin with gradient m. If the ray is y = 2x then
it will hit the point with coordinates x = 1 and y = 2 because these values satisfy
the equation. If the ray hits a lattice point with x = a and y = b the gradient m is
the fraction b/a. Consequently if m is not a genuine fraction (it may be √2, for
example) the light ray will miss all the lattice points.