There are no new ideas in Hollywood. The top 12 grossing films in the US last year were all, in one way or another, derivative content (6 super hero films, 3 remakes, 2 sequels, and a Star Wars). The increasingly serialized nature of the film industry may derive from the rise in recent years of television, a more episodic medium by nature. But films are also converging in structure.

Film writer Blake Snyder’s 2005 book Save the Cat (a great recap on the book and its influence is here), proposed a structure for writing films that follow a prescribed formula. (The title of the book stems from an archetypal scene, usually early in the film, in which the film’s hero saves a cat. The audience is then compelled to like and root for the hero. In Back to the Future, Marty McFly’s save the cat moment is being uber-cool on a skateboard.) Previously on this blog, we have discussed research on films showing how various low-level properties (like brightness and motion) correlate with this common narrative structure.

But films aren’t just similar to each other in content and narrative structure – they’re also similar to themselves! And this self-similarity has increased over the last 60 years or so. That is, according to a recent article in the Psychonomic Society’s journal Cognitive Research: Principles and Implications, James Cutting and colleagues show that certain properties of films are now more fractal-like than they have ever been.

First, for the uninitiated, let’s discuss fractals more generally. For many, the term fractals conjures up visualizations of the Mandelbrot set, a pattern that is made up of itself at smaller scales.

The Mandelbrot set reveals that fractals are self-similar: zooming in on one part of the image reveals a similar structure for the part and the whole. Turn this idea around, and it becomes clear that the larger elements (the large spirals above) occur less frequently, while smaller elements occur more frequently. In fact, this relationship between size (or some other non-spatial dimension) and frequency is a defining characteristic of fractals: 1/f where f = frequency.

In the time domain, shorter events occur more frequently than longer events. When 1/f frequency is generated for light, it results in a pinkish tint (because the longer wavelength red frequencies have higher power). The pink effect in light gives 1/f noise its name in sound: pink noise.

In the article by Cutting and colleagues, the researchers investigated a large set (almost 300) of films released from 1915 to 2015, to see if fractal patterns emerged during the course of cinematic history across seven different properties: shot duration, scene duration, motion, sound amplitude, luminance, clutter, and shot scale.

Cutting and colleagues mainly focus on shot duration, which we’ll use to illustrate the main findings. To discover whether shot duration exhibits fractal patterns, the researchers measured the duration of every shot in every film. For Back to the Future, that results in a graph like this:

Each point on this line graph is the duration of the first 512 shots of Back to the Future. Cutting and colleagues then fit a model to this graph, which estimates the amount of power for each frequency. (For those interested in the details, dive deep on the Fourier transform.)

The result, shown below for The Lion King, is an estimate of the power of each frequency.

If a film exhibits a fractal pattern in shot duration, as The Lion King does, this should result in a line with a slope near 1. More simply, this means that films have more shots of shorter lengths and fewer shots of longer lengths, occurring about as frequently as the inverse of their lengths.

Incredibly, shot duration in films has become more fractal-like over time. Using a type of 1/f estimator called an Exact Whittle Estimator, Cutting and colleagues do show an increase in the fractal quality of shot duration from 1915-2015, as shown in the graphs below.

The same trend toward pink noise can be seen in scene duration, motion, and sound amplitude (although not in luminance, clutter, or scale). Note, in the figure below, that sound amplitude fluctuations have decreased over time, and are now closer to pink noise, whereas shot duration, scene duration, and motion have increased over time, approaching pink noise from below.

Fractals are ubiquitous, occurring in coastlines, broccoli, mountain ranges, but also heartbeats, eye movements during visual search, and human cognition. In this last case, researchers found that when human subjects recreated temporal and spatial intervals, their errors were characterized by 1/f noise. Why fractals occur across these domains in human physiology is poorly understood – some suggest that fractals are emergent properties of complex systems more generally.

On this blog, we have discussed research on how filmmakers create shots and craft action to guide the audience’s attention. Cutting and colleagues propose that as filmmakers have become more gifted at capturing human attention from the inception of cinema in the 1920s through today, they have become better at creating stories that echo the pulse and rhythms of natural human life. In doing so, their films achieve something closer to the same 1/f noise in our own heartbeats, eye movements, and attentional patterns. Maybe that is what compels us to watch films, even as the stories are the same.

Reference for the article discussed in this post:
Cutting, J.E., DeLong, J.E., & Brunick, K.L. (2018). Temporal fractals in movies and mind. Cognitive Research: Principles and Implications. DOI:10.1186/s41235-018-0091-x.