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The Use of Mathematics to Describe Film Composition

Throughout these essays, I attempt to use some simple math, to analyze how filmmakers composed the images in their films. Throughout film history, many filmmakers have spent enormous care on how they compose the images in their films. Many spent hours on the set, carefully arranging the actors, the sets, the props and the camera, till everything is exactly right. Many others have also pre-planned, using story boards or sketches. Some, like Fritz Lang, had scale models of the sets built. Lang spent many hours every evening before a film shoot, carefully lining up all his shots for the next day, using these scale models: they look like doll houses.

John Ford told Peter Bogdanovich, summing up his talent: "The only thing I always had was an eye for composition." (See Bogdanovich's book, Who the Devil Made It?)

Many film viewers and critics believe they are seeing complex, beautiful compositions in the works of many great filmmakers.

None of the above is controversial or original. It is pretty widely accepted among filmmakers and viewers.

Talking about composition is precise detail is another matter. It is universally agreed that it is difficult to describe pictures in words. This is true whether they are paintings, photographs or films.

These articles attempt something new and different, at least for film study. They make a tentative attempt to use some simple geometry to analyze the compositions found in specific films. Frames from films are described in terms of circles, rectangles, planes and perpendicular lines.

This approach is tentative and experimental. It is not intended as the last word on composition. Virtually all film images are actually more complex than my analyses suggest. Still, it IS an attempt to talk in concrete terms about what is going on in the actual images of the films.

Art historians have used this approach. See, for example, the analysis of composition and proportion in "The Language of Painting" by Christopher Cornford, in Part 1, "Understanding Art" of The Illustrated Library of Art (1981) edited by David Piper. Cornford analyzes such paintings as Titian's Bacchus and Ariadne (1522-1523) and Raeburn's The Reverend Robert Walker Skating (circa 1784).

Orson Welles discusses and illustrates many grids in his Othello film in his documentary Filming Othello (1978).


Examples are in the following articles on specific films (click on the link to the director, then follow the bookmark at the top of the article to the mentioned film):

Camera Movement

Articles on the site frequently describe camera movement.

The following director articles discuss panning:

The following director articles discuss path / reverse path camera movement. The camera moves along a path; later it moves back along the same path, but in the reverse direction:

There is a detailed discussion of camera movement in the article on Joseph H. Lewis.

One might also want to check out my discussion of camera movement in the TV series Pacific Blue.

Some caveats:

I strongly suspect that great filmmakers did not explicitly use math on the sets of their films, while they were composing their shots. I doubt if Orson Welles or John Ford or Roberto Rossellini said "I need a circle here and a trapezoid perpendicular to the plane of the shot". Instead, my best guess is that they went through some sort of process of "creative visual thinking", in which they built up the images they wanted to see. Unfortunately, I do NOT know what that process felt like in the mind of Lang or Ford or Rossellini. I hope that someone will sit down with great contemporary masters of visual style, such as Curtis Harrington or Tran Anh Hung, and try to get them to explain "how they do it".

In addition, I do not know if thinking about composition in terms of geometrical concepts will be on the slightest use in training aspiring filmmakers. I am NOT advocating that students who are trying to learn how to make movies start thinking about their film composition in mathematical terms. This is an unexplored question. Some student filmmakers might get insight from reading a mathematical analysis. Others might regard it as an obstacle to the creative process of inventing beautiful images. To use an analogy, riding a bike can be analyzed in mathematical terms. This does not mean that kids learning to ride a bike need math to help them learn how to do it.

Testing these ideas

I have tried to make all of the descriptions in these articles concrete. Readers should be able to test all of them for accuracy. Any reader with a copy of the film, can compare the frames in the films to their analyses in the article. They can see for themselves whether the geometric descriptions are accurate or not.

Completeness is another question. Are the analyses picking up on the most important aspects of the image? Or are they leaving out key and all-important parts of the composition? This is more of a judgment call. Still, it is a concrete question that can be asked by readers of these articles, and one to which they can formulate concrete, discussible responses.

Math is not "Film Theory"

Mathematical ideas are certainly theoretical, with a small t. But they are not what many film professors today call Film Theory. This article is not the place to offer an exposition of Film Theory, a complex body of ideas and terminology advocated by some film professors to analyze movies.

I have never used Film Theory. Throughout all of these articles, one will find no Marxist, psychoanalytic, or Film Theory concepts whatsoever. I have never used any of these ideas, in my entire life. I regard films as works of art, and film makers as artists, in the full meaning of these terms.

For all my life, my political ideas have largely corresponded with those of the Democratic Party here in the United States. I have never been either a Communist or a conservative. You will never find any ideas in any of my writings that stem from Communism or conservatism. On the contrary, I have been a staunch opponent of Communism and conservatism throughout my entire life.