The light ray y = 2x hits the point A with coordinates x = 1 and y = 2 but it
will not strike the point B with coordinates x = 2 and y = 4 and all other points
‘behind’ A (such as C, with coordinates x = 3 and y = 6, and D with x = 4 and y
= 8) will be obscured. We could imagine ourselves at the origin identifying the
points that can be seen from there, and those that are obscured.
The points O ‘visible’ from the origin, and the obscured points ×
We can show that those points with coordinates with x = a and x = b which
can be seen are those that are relatively prime to each other. These are points
with coordinates, such as x = 2 and y = 3, where no number other than 1
divides both x and y. The points behind this one will be multiples, such as x = 4
and y = 6, or x = 6 and y = 9, and so on.
Pick’s theorem
The Austrian mathematician Georg Pick has two claims to fame. One is that he
was a close friend of Albert Einstein and proved instrumental in bringing the
young scientist to the German University in Prague in 1911. The other is that he
wrote a short paper, published in 1899, on ‘reticular’ geometry. From a lifelong
work covering a wide range of topics he is remembered for the captivating Pick’s
theorem – and what a theorem it is!