Even if (like me) you are one of those souls who are congenitally allergic to mathematics, so much of creating a beautiful image involves numerical principles. Many classic films have made use of the number known as the "Golden Ratio" (1.618), as well as the visual device known as One-Point Perspective (a way to make a two-dimensional plane look three-dimensional), and now Vimeo user Ali Shirazi has put together a visual essay on the use of these, and other visual principles, in Paul Thomas Anderson's There Will Be Blood, which was lensed by DP Robert Elswit.
Now, a bit of a primer:
The Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...) is a sequence of numbers that started as a way for 12th-century Italian mathematician Leonardo Fibonacci to calculate the breeding habits of rabbits (seriously). Each new number in the sequence is the sum of the two numbers before it, (i.e., 3+5=8, 13+8=21, etc.). Any two of these numbers is an approximation of the Golden Ratio, or 1.618, and the approximation gets more accurate the higher you progress in the sequence.
The Golden Ratio appears countless times in nature and art and is considered to be the ratio most pleasing to the human eye, though why this is so is a question that has never quite satisfactorily been answered. (Caveat: this is a vast oversimplification.)
This video explains the situation better than I ever could:
Many filmmakers are no doubt familiar with another compositional technique, the Rule of Thirds, which, simply defined, states that "an image is most pleasing when its subjects or regions are composed along imaginary lines which divide the image into thirds — both vertically and horizontally."
Single Point (or One Point) perspective is another visual device that has been used for thousands of years to give a three-dimensional quality to a two-dimensional work of art (whether painting, photograph, or motion picture), essentially by having lines radiate outward from a single vanishing point, giving the illusion of depth.
And now, Ali Shirazi has made a study of P.T. Anderson's There Will Be Blood, demonstrating how the film makes use of the Golden Ratio, Single-Point perspective, and tracking shots; it is a beautiful and educational visual analysis of the film, and a lesson for any filmmaker (or anyone) interested in the mathematics behind cinema. How are images constructed so as to be pleasing (or jarring, as the case may be) to the eye, and what is the numerical basis behind this phenomenon?
What do you think? Do you have any favorite examples from film (or other visual art) that can shed light on this fascinating phenomenon? Let us know in the comments!