Several years ago, I was trying to explain infinity to a fifth-grader class. In the story I told them, I was preparing to eat a piece of pie when a friend walked in on me. Out of courtesy I gave her half. Before I could eat my half, another friend came by, so I split it again. This happened over and over – in the story I have a lot of friends – and my snack kept getting smaller and smaller. How much pie would I have at the end of this? “No!” many of the fifth graders screamed. Together we wrote a sum on the board: ½ + ¼ + . . . We agreed that if you kept writing numbers forever, eventually they should add up to one. Then we talked about another infinite sum that had made its way across the internet: 1-2+3-4+. . . When I convinced them it was equal to ¼, I was pretty sure their murmurs of “mmm” weren’t dreams of cake.
A new documentary, “A Trip to Infinity,” attempts to convey this sense of wonder to Netflix’s massive audience. Populated by a diverse and captivating cast of mathematicians, physicists and a lost philosopher or two, the film, by Jonathan Halperin and Drew Takahashi, explores the infinite, with its puzzles and paradoxes, not only as a mathematical construct but also as an idea helps us calibrate the vastness of the universe and understand what it would mean if something went on forever, and forever, and forever. In images and interviews, the film considers whether there are physical manifestations of infinity, and whether it is possible for a mortal human to experience infinity.
Why are we so intrigued by the infinite? Perhaps it’s the tension between our finite lives and the seemingly limitless scope of our imagination, between the boundaries we experience and the potentially infinite universe we inhabit. Young people may think that life will last forever; older people, realizing that this is not the case, may look for a semblance of immortality in their inheritance. Buzz Lightyear teaches kids in ‘Toy Story’ that life is full of infinite possibilities; Hamlet, lamenting the finiteness of life, remembers Yorick as a man of endless jokes. Perhaps understanding infinity, even just a little bit, is a way to feel some control and comfort in the face of life’s great questions.
Mathematical documentaries are always a challenge for filmmakers because mathematics 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 sequence of talking heads? One of the best answers to that question is “Donald in Mathmagic Land” (1959), which was part of a series of science and education documentaries that Disney produced in the 1950s and 1960s. In less than thirty minutes, the film takes the viewer — and Donald Duck — from the ancient Greeks to futuristic astronauts, introducing concepts like number theory and geometry. Crucially, “Mathmagic Land” combines fun and fact without making the math too simplistic and without talking to the viewer. Even when animated, “Mathmagic Land” is helpful light on metaphor. So is “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 mathematical magic. For example, it includes a cartoon called “The Infinite Hotel,” based on a thought experiment by twentieth-century German mathematician David Hilbert. In a voice-over by the mathematician Steven Strogatz, we hear that the hotel is occupied, but that it can always accommodate more guests – even an infinite number of new guests. Strogatz explains the infinite sum that robbed me of my piece of pie, ½ + ¼ + ⅛ . . ., by describing a hotel manager who has a limited amount of time to prepare rooms for new arrivals. The film also succeeds when it trusts its captivating experts—an all-star cast that includes physicists Janna Levin, Stephon Alexander and Carlo Rovelli, and philosopher Rebecca Goldstein—to find their own words for what infinity is and what it makes. endlessly interesting.
The movie goes awry, I think, when it tries to dress up math in psychedelic animation, and when it asks math and science experts to engage in armchair philosophy. As a viewer, I often felt that I had been invited to an epistemological fishing expedition. In an awkward order, participants are handed a small black orb to ponder, then instructed, “Tell me what you’re thinking about with infinity.” You can almost feel them squirming as they try to answer in front of the camera. A closing question – “Do you think human creativity is infinite?” – similarly calls for an edit. At other times, the expert voiceovers are coupled with animations of nesting circles and tiled spaces, the kind of gimmicks that have made science fiction movies clichés. The most penetrating animation is a train that interrupts the mathematician Moon Duchin twice as he ponders what it would mean for a mathematical object like infinity to ‘exist’. The second appearance of the train completely blocks her out of sight and thunders over her mind, as if the ideas behind it aren’t interesting enough in themselves. As a mathematician I may be biased, but I think I am.
Is the universe as infinite as we can imagine? We may never know, but the reasons for that are fascinating in their own right. The surprise, the film points out, is that even a finite universe can seem infinite to its inhabitants. To clarify this, the viewer goes on a journey to a four-dimensional world. While some mathematicians claim that they can visualize things in four dimensions, most of us can only work by analogy here: imagine that you are a dot that can only follow the path of a circle on a piece of paper. If you only experience one dimension, you might think that your universe continues in a straight line forever. The same goes for a dot that can wander around the two-dimensional surface of a billiard ball: you might think that your world is infinite in all directions, even though we three-dimensional beings can look at the paper or billiard ball and recognize that each has its limits. In one of the film’s best scenes, astrophysicist Delilah Gates, speaking straight into the camera, contemplates whether our universe could be the three-dimensional equivalent of these finite spaces. You don’t need 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 felt limited at the time; I was unhappy with college and not inspired by the chemistry labs I wanted to spend my time on. One day I walked into my campus bookstore and pulled a math book off the shelf. I found it so fascinating that I sat down in the aisle, oblivious to the shoppers shuffling around me. The simple act of counting, I read, could be a source of surprise. For example, there are as many counting numbers as there are even numbers, even though that would seem to imply that two times infinity is exactly the same as infinity. There are as many fractions as numbers, even though an infinite number of fractions can fit between two numbers. How can this be? I’ve learned that there is a greater infinity, the one you get when you try to count all the numbers that can be written as decimals. This last set is so infinite that it is uncountable: however you try to list these numbers, you will necessarily omit some—in fact an infinite number of numbers. That 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 gateway to a wonderful world. I’m still here.