Curlicue Picture

We see pictures of the golden curlicues. This is the random walk in the complex plane, where the individual steps are exp(2Pi i a k2), where a is the golden ratio. This has relations with the golden graph on a formal level because the infinite sum for a in the upper half plane can be written as a Birkhoff sum.

The general curlicue problem deals with the case exp(i a k + i b k2), where a,b are real numbers. It includes the problem to drive in the desert as follows: look up the big hand of your watch, drive in that direction for a specific fixed time T then look up the direction of the big hand, then drive in that direction for time T, then look up the big hand again, drive in that direction etc. The following pictures show how your path can look like when driving for longer and longer time.