In March 2023, Calcea Johnson and Ne'Kiya Jackson, two high school students, proved the Pythagorean theorem in a way that was previously thought to be impossible: using trigonometry. In the 2,000 years since trigonometry was discovered, it has always been assumed that any alleged proof of the Pythagorean theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis), the author states that "There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem." However, that isn't entirely accurate. The authors came up with a new proof of the Pythagorean theorem that is based on a fundamental result in trigonometry—the Law of Sines—and they demonstrate that the proof is independent of the Pythagorean identity sin²x + cos²x = 1.

Wilson's theorem: p is a prime number if and only if (p – 1)! + 1 is divisible by p

Johann Bernoulli was born exactly 358 years ago today. In 1696, he posed the famous Brachistochrone problem: What curve lets a particle slide - without friction and under gravity - from one point to another in the least time? Newton solved it in 1697

For any prime number p ≥ 5 , p²-1 is a multiple of 24

Ramanujan said that he received his formulas from God

Human-like beings are doomed to drop toast butter-side down 🥪 In 1995, Robert Matthews wrote a paper showing that toast landing butter-side down isn’t just bad luck or evidence of Murphy’s Law ('If it can go wrong, it will') - it’s physics and ultimately ascribable to the values of the fundamental constants. Toast typically begins to rotate as it tips off a table, but the height of most tables (~75 cm) is just enough for it to rotate about half a turn, landing butter-side down. This isn’t due to butter's weight or aerodynamics, which are negligible. Rather, it's about the torque and time during the fall. In this paper, Matthews builds a detailed dynamical model of the toast’s tumble and shows experimentally that under realistic conditions (with little horizontal velocity), there's a built-in bias toward a butter-side down landing. Surprisingly, this bias is linked to fundamental constants: table height is constrained by human height, which in turn is limited by biomechanical stability and molecular bond strength (à la Press, 1980). Given the values of fundamental constants, the result is universal - all intelligent, human-like beings are doomed to drop toast butter-side down. To avoid this fate? A little push adds horizontal velocity, reducing rotational torque, a counterintuitive fix but backed by physics.

Paul Erdös on the infinitude of Mathematics

In 1960, Freeman Dyson proposed a megastructure to capture all the Sun’s energy - the Dyson Sphere. That’s what a Type II civilization needs. At ~2.5% annual growth (like 1960–2025), we’ll need a whole star in 1,200 years. A Dyson Sphere isn’t sci-fi - it’s a forecast.

The fifth hyperfactorial: 5⁵ × 4⁴ × 3³ × 2² × 1¹ = 86400000 milliseconds is exactly 1 day!

Happy 50th birthday Terence Tao! 🎉