Trails of the Tail

by Simon Whitechapel

This is the pattern created when each of five dogs at the vertices of a pentagon begins to chase the tail of the dog one place ahead of it, i.e. chase(5, n + 1):

I call this pattern a ourogram, from Greek ουρον, ouron, “tail” + γραμμα, gramma, “symbol”. Here’s the ourogram when each of eight dogs at the vertices of an octagon chases the tail of the dog three places ahead of it, i.e. chase(8, n + 3):

(n + 3 mod v is returned, i.e. the remainder when n + 3 is divided by the number of dogs, but with 8 mod 8 = 0 = 8)

Here’s the ourogram for chase(8, 2n) — dog one (in the one o’clock position) chases dog two, two chases four, three chases six, four chases eight, five chases two, eight chases itself, and so on.