Studying notions of long-range order in symbolic dynamical systems via dynamical invariants has a long history which goes back to the seminal work of Hedlund and Morse. We will provide an overview of the low-complexity concepts of mean equicontinuity and amorphic complexity, and examine how they are interrelated. We will then study amorphic complexity, tameness, and nulness in the class of automatic systems---systems arising from constant length substitutions. We will provide a closed formula for the amorphic complexity of any automatic system and further show that tameness or nullness of such systems can be characterized succinctly through amorphic complexity: an infinite minimal automatic system is tame if and only if it is null, which occurs precisely when its amorphic complexity is one. In the proof we will see methods from fractal geometry and some new dynamically-defined pseudometrics.