IFS and Film

In his project for the autumn, 2000, fractal geometry course, Olaf Schneider used driven IFS to study patterns in the film "Pulp Fiction." Schneider selected this film because of its merits, established through the many awards it won, and the way it polarized audiences. People seemed to either love or hate the film, leading to Roger Ebert's remark that "it is possibly the most unpopular movie ever to gross $100 million at the American box office." Schneider gives a good explanation for his choice of this film.

"'Pulp Fiction' delights some audience members and disturbs others, I think, for the same reason: because it toys with their expectations. The movie is more subtle and complex than at first it seems. The screenplay turns out to contain the answers to mysteries that baffle viewers in a first viewing, and it makes connections that only occur to you after watching the movie several times. The film tells interlocking stories, which unfold out of chronological order, so that the movie's ending hooks up with the beginning, most of its middle happens after the ending , and a major character is onscreen after he has been shot dead. This seemingly circular plot first led me to suspect that I might find a fractal pattern in this movie."

He continues, "many things in nature produce fractal patterns, perhaps we as humans identify them on an unconscious level and this unconscious awareness allows us to reproduce these patterns in our attempt to recreate life through films. The rhythm of a movie then, might consist of a correlation of several visual features that form a pattern that corresponds to fractals in nature."

Schneider performed two driven IFS experiments, both based on the sequence of shots in the film.

The first experiment involved the distance of the shots.
bin shot purpose
1 extreme long and long show the spatial relations among the important figures
2 medium and medium close-up show two characters in conversation
3 close-up shows facial features during conversations
4 extreme close-up final shot in a build-up of tension, or part of a fast action sequence

Here is the IFS of "Pulp Fiction," driven by shot distance. (The triangles on hte right will be explained in a moment.

The faint gasket appearing results from the relatively low number of extreme close-ups (bin 4), about 6%. Frequent combinations of close-ups and medium close-ups (bins 2 and 3) explain the strong diagonal. Schneider speculates that in more standard movies the squares 141 and 1431 would contain no points because they represent juxtapositions of extreme close-ups and extreme long-shots. The interlocking plot lines in "Pulp Ficution" validate these combinations.

Looking at the diagonal led Schneider to an interesting observation. Although the points are not even approximately symmetrically distributed across the diagonal, some empty traingles are. These, too, he related to the jerkiness of 14 and 41 combinations. However, the situation is complicated by the presece of some points in the squares 14 and 41. The empty triangle in square 41 is made up of squares 411, 4131, 4121, and so on. Speculations about these squares leads to the question of whether they are empty because of a true exclusion, or because not enough data is present. Of course, we cannot add more shots to a film (Schneider counted 1029 in "Plup Fiction"), but we can exercise some caution about how small an empty square we trust. For instance, there are 46 = 4096 squares with length 6 address. With only 1029 points in the plot, most length 6 address squares will be empty, whether or not the address represents an impossible combination.

Schneider's second experiment involved the duration of the shots.
bin duration purpose
1 1-4 seconds rapid action
2 5-8 seconds informationon how the environment has changed after a fast sequence
3 9-30 seconds plot development
4 > 30 seconds major plot points

Again, we see a faint gasket, and again it is caused by a relatively small number of 4s (long takes are rare), only about 4.3% of all points. Further, the square 14 is less populated than 11, 12, and 13. Is this a consequence of the smaller number of 4s, or of the unlikeliness of the 41 combination? The percentage of points of square 1 in subsquare 14 is about the same as the percentage of all points in square 4. In general, Schneider observed that fewer empty squares suggests there little relation between previous takes and the current take. Why this should be true is unclear.

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