Category: Mathematics

The sum of the reciprocals of the natural numbers diverges, but slowly, like the logarithm of the number of terms. The sum of the reciprocals of the prime numbers also diverges, but even more slowly, like the logarithm of the logarithm of the number of terms, as the primes are sparse in the naturals! Here…

When searching for prime numbers, the next prime number is no larger than twice the current number. Postulated by Joseph Bertrand, first proved by Pafnuty Chebyshev, I present an elementary proof based on one by the teenage Paul Erdős. Erdős was one of the most prolific twentieth century mathematicians, publishing about 1500 articles with more…

Just a couple of months after announcing the remarkable discovery of a single shape that forces a non-periodic tiling of the plane, Smith, Myers, Kaplan, and Goodman-Strauss have announced an improved aperiodic monotile or ein stein. (Ein stein is “one stone” in German.) The hat and turtle shapes tile the plane only non-periodically, but with their…

This academic year has been thrilling: first nuclear fusion breakeven, now an ein stein! Last week, a preprint at arxiv.org by David Smith et al. announced an “ein stein”, or one stone, a shape that forces a non periodic tiling of the plane, ending a half-century quest by many researchers, including me. A retiree and…

On 1852 October 23, Francis Guthrie noticed that he needed only 4 colors to color the counties of England so no two bordering counties shared the same color. This works for any map, but only in 1976, and with the aid of a computer, did Kenneth Appel and Wolfgang Haken finally prove the 4-color theorem. Here I outline…

Slide rules were the analog computers that ruled science and engineering for 400 years. Their brilliant innovation was using logarithms to convert multiplication and division to addition and subtraction, and Slide rules feature logarithmic scales that slide past each other. For straight slide rules, logarithms of the numbers are proportional to their distances along them.…

In four dimensions, you can’t tie your shoelaces — because 4D knots don’t work. Any 1D curve in 4D space can be continuously deformed to the unit circle, which is an unknot. The looping animation below demonstrates how to undo a trefoil knot in 4D, where rainbow colors code the 4th dimension. The animation pauses when…