The Fly, or the Swallow
Today Hacker News carried Paul Halmos's 1973 essay "The Legend of John von Neumann". In it two cyclists ride towards each other while a fly shuttles between their front wheels, and von Neumann, asked how far the fly flies, answers at once by summing the infinite series. Eugene Wigner told the same story on film in 1966 with other numbers and a different bird. Wikipedia blends the two. Below the fly flies every leg, and you can see where a real one would be squashed.
| Leg | Flown this leg | Still to fly | Leg lasts |
|---|
The view zooms with every leg so the gap between the front wheels always fills it; the scale written under the road keeps shrinking. The bicycles are not to scale. The flier is drawn at true scale against that gap once it is big enough to see, and as a dot before that. It is squashed when the gap closes to its own length, taken as 6.5 mm for a housefly and 18 cm for a barn swallow.
Four tellings
| Who | Riders | Apart | Each at | Flier | Answer | Von Neumann says |
|---|---|---|---|---|---|---|
| Wigner, on film, 1966 | two cyclists | 40 miles | 20 mph | a swallow, 50 mph | 50 miles | "a simple infinite geometrical series"; the asker was Max Born |
| Halmos, American Mathematical Monthly, 1973 | two bicyclists | 20 miles | 10 mph | a fly, 15 mph | 15 miles | "all I did was sum the infinite series" |
| MathWorld | two trains | 100 km | 50 km/h | a fly, 75 km/h | 75 km | adds that von Neumann's version had "a fly and two bicycles" |
| Wikipedia, John von Neumann, today | two trains, "in the original variant" | 20 miles | 10 mph | a fly, 15 mph | 15 miles | "All I did was sum the geometric series", cited to Halmos pp. 386–387 |
Wikipedia's quotation has Halmos's sentence with Wigner's adjective. Halmos wrote "infinite"; "geometrical" is Wigner's word, from the film. Its "original variant" uses Halmos's numbers with MathWorld's trains, but neither of the two tellings it cites puts trains in von Neumann's story. Both men who knew him put the riders on bicycles.
Wigner's words I checked by transcribing the film's soundtrack, 16:50 to 19:20 of the Mathematical Association of America's 1966 documentary, with two different speech models. They agreed on every number and on the phrase.
What a real fly would manage
The series has infinitely many terms, but a fly is not a point. In Halmos's version each leg is a fifth of the one before. After nine legs the front wheels are 16.5 mm apart, and during the tenth the gap closes to a housefly's 6–7 mm. The fly has flown 15 miles less about 5 mm. The whole tenth leg would take a millisecond and a half, and the fly gets 1.1 ms into it.
Wigner's swallow shrinks each leg only to three-sevenths, so it gets fifteen full legs before the gap reaches its 17–19 cm on the sixteenth. It falls 22 cm short of 50 miles. It also flies at 50 mph, or 22 m/s. Wikipedia's barn swallow article gives that bird about 11 m/s, up to 20.
One more small thing. Halmos sets the fly off from the southbound bicycle towards the northbound one, then calls this "the first, northbound, leg". The fly is heading south. He probably means the leg to the northbound bicycle.
What stands on what
| Item | Source |
|---|---|
| Halmos's telling and quotation | P. R. Halmos, "The Legend of John von Neumann", American Mathematical Monthly 80 (1973), pp. 386–387, the PDF linked from Hacker News |
| Wigner's telling | John von Neumann – A Documentary (Mathematical Association of America, 1966), from 16:50: my transcripts, parakeet and Whisper large-v3-turbo |
| Trains, 100 km | Wolfram MathWorld, "Two Trains Puzzle" |
| Wikipedia's wording | Wikipedia, John von Neumann, read today |
| Housefly 6–7 mm; barn swallow 17–19 cm, about 11 m/s, up to 20 m/s | Wikipedia, Housefly and Barn swallow (the speeds citing Liechti and Bruderer) |
| Legs, gaps, shortfalls | My arithmetic, also run in the page's script |