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"A journey to infinity" and the delicate art of mathematical documentary

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Several years ago, I tried to explain infinity to a class of CM2 students. In the story I told them, I was about to eat a piece of cake when a friend surprised me. Out of politeness, I gave him half. Before I could eat my half, another friend came over, so I shared it again. It happened over and over again – in the story, I have a lot of friends – and my snack kept shrinking. How much cake will I have at the end? “None!” many fifth graders shouted. Together we wrote a sum on the board: ½ + ¼ + . . . We agreed that, eventually, if you kept writing numbers forever, they had to add up to one. Next, we talked about a different infinite sum that had made its way onto the internet: 1-2+3-4+ . . . When I convinced them that was ¼, I was pretty sure their whispers of “mmm” weren’t cake dreams.

A new documentary, “A Trip to Infinity,” attempts to convey that sense of wonder to Netflix’s massive audience. Populated by a diverse and engaging cast of mathematicians, physicists and a wandering philosopher or two, the film, by Jonathan Halperin and Drew Takahashi, explores infinity, with its puzzles and paradoxes, not just as a construction mathematical but also as an idea. it helps us calibrate the vastness of the universe and grasp what it would mean if something went on forever, and forever, and forever. In footage and interviews, the film questions whether there are physical manifestations of infinity, and whether it’s possible for a mortal person to experience infinity.

Why are we so intrigued by infinity? Perhaps it’s because of the tension between our finite lives and the seemingly limitless range of our imaginations, between the limits we know and the possibly infinite universe we inhabit. Young people may think that life will last forever; the elderly, realizing that this will not be the case, might seek some semblance of immortality in their inheritance. Buzz Lightyear, in “Toy Story”, teaches children that life is full of endless possibilities; Hamlet, while lamenting the finitude of life, remembers Yorick as a man of endless jokes. Perhaps making sense of infinity, even a little, is a way to feel some control and comfort in the face of life’s big questions.

Math documentaries always pose a challenge to filmmakers because math does not exist in the realm of images but in the realm of ideas. How do you illustrate a complicated concept without resorting to gimmicks and distractions, or limiting your film to a talking head sequence? One of the best answers to this question is “Donald in Mathmagic Land” (1959), which was part of a series of science education documentaries Disney produced in the 1950s and 1960s. In less than thirty minutes, the film takes the viewer – and Donald Duck – from ancient Greeks to futuristic astronauts, introducing concepts such as number theory and geometry. Basically, “Mathmagic Land” combines fun and reality without oversimplifying the math and talking to the viewer. Even though it’s animated, “Mathmagic Land” is helpfully light on metaphor. The same goes for “The Proof” (1997), the popular “Nova” documentary that captured the excitement of Andrew Wiles’ proof of Fermat’s Last Theorem.

“A Trip to Infinity” has moments of math magic. It includes, for example, a cartoon titled “The Infinite Hotel”, based on a thought experiment by 20th-century German mathematician David Hilbert. In a voice-over by mathematician Steven Strogatz, we learn that the hotel is busy, but that it can always accommodate more customers, even an infinity of new customers. Strogatz explains to me the infinite sum that deprived me of my piece of cake, ½ + ¼ + ⅛. . ., describing a hotel manager who has limited time to prepare rooms for new arrivals. The film also succeeds when it trusts its engaging experts – an all-star cast that includes physicists Janna Levin, Stephon Alexander and Carlo Rovelli, and philosopher Rebecca Goldstein – to find their own words on what infinity is. and what makes it infinitely interesting.

The movie goes awry, I think, when it tries to disguise math in psychedelic animations, and when it asks experts in math and science to engage in armchair philosophy. As a spectator, I often had the impression of being invited to an epistemological fishing. In an awkward sequence, participants are given a small black orb to gaze upon, then given the instruction, “Tell me what this makes you think of infinity.” You can almost feel them squirming as they try to answer for the camera. One last question – “Do you think human creativity is infinite?” – also requires modification. At other times, expert voiceovers are paired with animations of interlocking circles and tiled spaces, the kind of gimmicks sci-fi movies have turned into cliches. The most intrusive animation is a train that twice interrupts mathematician Moon Duchin, who ponders what it would mean for a mathematical object like infinity to “exist.” The second appearance of the train completely blinds him and rumbles in his thoughts, as if the underlying ideas weren’t interesting enough on their own. As a mathematician, I may be biased, but I think they are.

Is the universe as infinite as we might imagine? We may never know, but the reasons are fascinating in themselves. The surprise, the film points out, is that even a finite universe can seem infinite to its inhabitants. To explain, the viewer goes on a journey through a four-dimensional world. While some mathematicians claim the ability to visualize things in four dimensions, most of us can only proceed here by analogy: Imagine you are a point that can only follow the path of a circle on a sheet of paper. If you experience only one dimension, you might think that your universe goes on forever in a straight line. The same goes for a point that can wander around the two-dimensional surface of a billiard ball: you might think your world is infinite in all directions, even though we three-dimensional beings can look at the paper or the billiard ball. and recognize that everyone has their limits. In one of the film’s best scenes, astrophysicist Delilah Gates, speaking directly to the camera, wonders if our universe could be the three-dimensional equivalent of these finite spaces. No need for animation to appreciate it. These kind of human moments are the best reason to watch the film.

When I was a freshman in college, I had the kind of epiphany that “A Trip to Infinity” hopes to inspire. My life, at the time, felt limited; I was unhappy with college and uninspired by the chemistry labs I had planned to spend my time in. One day I walked into my campus bookstore and pulled out a math book from the shelf. I found it so captivating that I sat in the aisle oblivious to the shoppers who had to move around me. The simple act of counting, I read, could be a source of surprise. There are as many numbers counted as even numbers, for example, even though this seems to imply that twice infinity is exactly the same as infinity. There are as many fractions as numbers counted, even though an infinite number of fractions can fit between two numbers counted. How is it possible ? I learned that there is a greater infinity, the one you get by trying to count all the numbers that can be written in decimal form. This last collection is so infinite that it is uncountable: however you try to enumerate these numbers, you will necessarily leave out a few, in fact, an infinity of numbers. This book, a little worn around the edges and full of my marginalia, is in front of me as I type this. For me it was a portal to a wonderful world. I’m still there. ♦

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