Several years ago, I tried to explain infinity to a fifth grade class. In the story I told you, I was getting ready to eat a piece of cake when a friend found me. Being courteous, I gave him help. Before I could eat my half, another friend came over, so I split it again. This happened over and over again (in the story, I have a lot of friends) and my snack kept getting smaller. How much cake will I have at the end of this? “None!” many of the fifth graders screamed. Together, we write a sum on the board: ½ + ¼ + . . . We agreed that eventually, if you kept writing numbers forever, they had to add one. We then talked about a different infinite sum that had been making its way onto the internet: 1-2+3-4+. . . When I convinced them that it equaled ¼, I was pretty sure their mutterings of “mmm” weren’t cake dreams.
A new documentary, “A Voyage to Infinity,” attempts to convey this sense of wonder to Netflix’s huge audience. Featuring a diverse and engaging cast of mathematicians, physicists and a stray philosopher or two, the film, by Jonathan Halperin and Drew Takahashi, explores infinity, with its puzzles and paradoxes, not only as a mathematical construct but also as an idea. that helps us gauge the immensity of the universe and understand what it would mean for something to continue for ever and ever. In footage and interviews, the film contemplates whether physical manifestations of infinity exist and whether it is possible for a mortal person to experience infinity.
Why are we so intrigued by infinity? Perhaps it is because of the tension between our finite lives and the seemingly limitless range of our imaginations, between the limits we experience and the possibly infinite universe we inhabit. Young people may think that life will last forever; older people, realizing that it won’t be, might look to their legacies for a semblance of immortality. 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 infinite 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.
Mathematical documentaries always represent 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 tricks 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 education documentaries that 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. Crucially, “Mathmagic Land” blends fun and fact without oversimplifying the math and without speaking ill of the viewer. Even if animated, “Mathmagic Land” is helpfully light on metaphors. So is “The Proof” (1997), the popular documentary “Nova” that captured the excitement of Andrew Wiles’s proof of Fermat’s Last Theorem.
“A Trip to Infinity” has moments of mathematical magic. It includes, for example, a cartoon called “The Infinite Hotel,” based on a thought experiment by the 20th-century German mathematician David Hilbert. In a voiceover from mathematician Steven Strogatz, we learn that the hotel is busy, but can always accommodate more guests, even a myriad of new guests. Strogatz explains the infinite sum that deprived me of my slice of cake, ½ + ¼ + ⅛. . ., describing a hotel manager who has a finite amount of time to prepare rooms for new arrivals. The film also succeeds when it relies on its likeable pundits, an all-star cast that includes physicists Janna Levin, Stephon Alexander and Carlo Rovelli, and philosopher Rebecca Goldstein, to come up with their own words on what infinity is and what makes it infinity. infinitely interesting.
I think the movie goes awry when it tries to dress up math with psychedelic animations and when it asks math and science experts to engage in classroom philosophy. As a viewer, I often felt as if I had been invited on an epistemological fishing expedition. In an awkward sequence, participants are handed a small black orb to gaze at and are then instructed, “Tell me what infinity makes you think of.” One can almost feel them squirm as they try to answer for the camera. A final question, “Do you think human creativity is infinite?”, also cries out for editing. At other times, expert voiceovers are combined with animations of nesting circles and tiled spaces, the kind of tricks that science-fiction movies have become 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 train’s second appearance completely blocks her from view and rumbles over 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 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 appear infinite to its inhabitants. To explain, the viewer embarks on a journey into a four-dimensional world. While some mathematicians claim to have the ability to visualize things in four dimensions, most of us can only proceed here by analogy: Imagine that you are a point that can only follow the path of a circle on a piece of paper. If you experience only one dimension, you might think that your universe goes on forever in a straight line. The same is true for a point that can move 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 limits. In one of the film’s best scenes, astrophysicist Delilah Gates, speaking directly into the camera, contemplates whether our universe could be the three-dimensional equivalent of these finite spaces. You don’t need an animation to appreciate it. These kinds of human moments are the best reason to watch the movie.
When I was a freshman in college, I had the kind of epiphany that “Journey to Infinity” hopes to inspire. My life, at the time, felt limited; I was unhappy with college and uninspired by the chemistry labs where I had planned to spend my time. One day, I walked into my campus bookstore and pulled a math book off the shelf. I found it so fascinating that I sat in the aisle, oblivious to the shoppers who had to crawl around me. The simple act of counting, I read, could be cause for surprise. There are as many numbers to count as even numbers, for example, although that seems to imply that twice infinity is exactly the same as infinity. There are as many fractions as there are counting numbers, although an infinite number of fractions can fit between two counting numbers. How could this be? I 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 collection is so infinite that it cannot be counted: no matter how you try to enumerate these numbers, you will necessarily leave some, in fact, an infinity of numbers. That book, a little worn around the edges and full of my margins, is in front of me as I write this. For me, it was a portal to a world of wonder. i’m still there ♦