It’s a popular recreation to try to arrange four 4s into various expressions to generate the whole numbers, like so:

1 = 4 ÷ 4 + 4 – 4
2 = 4 – (4 + 4) ÷ 4
3 = (4 × 4 – 4) ÷ 4
4 = 4 + 4 × (4 – 4)
5 = (4 × 4 + 4) ÷ 4

In 1881 a writer to the London journal Knowledge noted that each of the first 20 integers except 19 can be generated using the operations +, -, ×, and ÷. In 1964 Martin Gardner found that if you use square roots, decimals, factorials, concatenations (444), and overline (.444 …) then every positive integer less than 113 becomes possible. (113 is surprisingly hard; it becomes possible if you use percents or the gamma function.)

In 2001 a team of mathematicians from Harvey Mudd College found that you can even get four 4s to approximate some notable constants if you use a whip and a chair:

That expression for e is accurate to 21 decimal places; it can be made arbitrarily accurate by repeatedly replacing 4 with 4!. The authors note that similar expressions can be derived using three 3s or five 5s.

Amazingly, they also approximated g, the acceleration due to gravity, with four 4s, as well as Avogadro’s number N_A.

(A. Bliss, S. Haas, J. Rouse and G. Thatte, “Math Bite: Four Constants in Four 4s,” Mathematics Magazine 74:4 [October 2001], 272.)

In Southeast Asia, fireflies synchronize their flashing. Observing them in Siam in the 1920s, naturalist Hugh Smith wrote, “Imagine a tenth of a mile of river front with an unbroken line of [mangrove] trees with fireflies on every leaf flashing in synchronism. … Then, if one’s imagination is sufficiently vivid, he may form some conception of this amazing spectacle.”

The phenomenon was so unexpected that some initially dismissed the reports as an illusion; Phillip Laurent “could hardly believe [his] own eyes, for such a thing to occur among insects is certainly contrary to all natural laws.”

Each male fly’s flashes are initially sporadic, but they adjust their timing according to those around them until they’re synchronized. This helps identify them to females of their own species. Biologist John Buck observed, “Centers of synchrony built up slowly, two individuals often flashing independently for up to half a minute (about fifty cycles) before the flashes coincided. At this point their rhythms locked together and continued in synchrony thereafter.”

In 2015 Robin Meier and Andre Gwerder used LEDs to artificially direct the speed and rhythm of thousands of flashing fireflies (above), using this technique to “explore the idea of free will and transform a machine into a living actor inside a colony of insects.”

(Ying Zhou, Walter Gall, and Karen Nabb, “Synchronizing Fireflies,” College Mathematics Journal 37:3 [May 2006], 187-193.)