Received this beautiful diagram in my email today:
The obvious question is, of course, whether the values can be made exact. It turns out that the answer is yes. Precisely:
Suppose distinct positive reals x, y, z, u are the values of four resistors that are self-replicating in the following sense: There exist four electrical networks, each of which is made using the four resistors (once and only once), and the resistances of the four networks are x, y, z, u. What are the values of x, y, z, u?
The solution is unique up to some obvious symmetries (such as scaling). It is known that there is no solution for 2 or 3 resistors.
Source: Problem of the Week by Stan Wagon, Macalester College.