"Does this famous dialogue ring a bell? ‘You said one out of fourteen million, we’d win, yeah? Tell me this is it.’"
The above quote is a classic example of the use of probability in the movie, Avengers: Endgame. This is just one scenario where mathematics is used as a guiding tool in the movie’s narrative.
Do you know that scriptwriters use graph theory to help create believable and captivating relationships between characters in a film? The scriptwriter creates a graph that represents the relationships between characters in a film. Each character is like a little dot, and the connections between them are like lines.
This is one perspective to depict the relationship between mathematics and cinema. There are many such uses of maths in the narrative of many movie plots.
There was a famous psychological thriller film that was released in 1998. The critically acclaimed movie is named after the mathematical constant ‘pi’. The plot focuses on a mathematician obsessed with establishing complete order in the real world. He contrasts two seemingly irreconcilable entities: the imperfect irrationality of humanity and the rigour and regularity of mathematics, specifically number theory. The film explores themes of religion, mysticism, and the relationship of the universe to mathematics.
A plethora of similar movies piques a newfound appreciation for mathematics through its out-of-the-box story modelling.
Did you know that the Lord of the Rings trilogy used 263 special effects artists, all of whom used a vast amount of maths, to make them perfect?
Characters like the Minions are described mathematically as a collection of points, curves, and surfaces. These points are moved around to give the impression of a living and breathing minion, interacting with its environment.
Creating animated films generally starts with artists sketching thousands of hand-drawn characters and scenes using pen and paper or with the help of computer software like Photoshop. These blueprints transfer into animating programs where they are brought to life by underlying math, including geometric coordinates (to map a character’s place in space using an X and Y axis graph), trigonometry (to move characters), linear algebra (to show the way that an object is rotated and shifted and made larger and smaller), and even integral calculus (to simulate the movement of lighting accurately).
While simple polygons such as triangles are useful to define the shape of an object, they do not produce realistic images. To develop a smoother image, subdivision algorithms are used to iteratively add more points.
While simple polygons such as triangles are useful to define the shape of an object, they do not produce realistic images. To develop a smoother image, subdivision algorithms are used to iteratively add more points
We see images only because of the light that is reflected from them. Different surfaces reflect light in different ways. Light from these object surfaces reaches our eyes from many different directions. This complex process has to be reproduced correctly for realistic representation. The procedure for this takes up a great deal of computer time and energy.
Fun Fact: In a James Bond movie production, the above process utilised so much power that there was a power cut in the London District of SOHO where the movie was being produced.
Typically, millions of mathematical equations are solved for every frame of an animated film.
The head of research at Pixar was recently asked: ‘What advice would you give someone who wanted to use math to make movies?’
His response: ‘Learn as much mathematics as you can, particularly applied math. The areas of mathematics we use most heavily today are Euclidean and affine geometry, trigonometry, linear algebra, calculus, and numerical analysis. We don’t know what the mathematical tools of tomorrow might be, so we’re counting on the next generation of employees to tell us.’
This quote underlines that mathematics behind the movies is undoubtedly one of the most dynamic and growing applications of modern mathematics, with many exciting possibilities for the future.
Film equipment is built on the principles of science and technology, which are, in turn, based on mathematical principles. The simplest video camera requires a deep understanding of optics, geometry, and trigonometry.
Every shot in a movie is planned out for its artistic angle. It is also calculated for the perfect lighting calibration and digital sound levels. The video camera plays a large part in this process, which requires math to determine the best aperture speeds, focal points, and camera angles.
Film budgets are complicated. Costs to be considered range from actors, sets, and special effects to post-production costs. To ensure restraint in finances, producers and budgeters use mathematical theories and techniques.
Two commonly used ones are linear programming and cost-benefit analysis. Linear programming optimizes resources and minimizes costs. It determines the best way to allocate resources for a particular film. Cost-benefit analysis considers the benefits and costs of a decision. Example: A film budgeter deduces the best way to allocate resources for special effects while keeping in mind both the cost of using higher-quality special effects and the benefits of using them.
Every aspect of film and filmmaking relies on math. It is involved intricately in every step of the movie-making process from the script to the design of the video equipment filmmakers use, to the production of the film itself. In conclusion, the magic of any movie lies in the math of it.
