Until 2022, every known tiling of the plane using tiles of the same shape eventually repeated itself. Then amateur mathematician David Smith discovered a remarkable tile that will cover an infinite plane but only in a nonperiodic way.

This solves an open problem in mathematics — for years researchers had been seeking an aperiodic monotile, or “einstein,” from the German for “one stone.”

Technically Smith’s tile, known as the “hat,” must be used in combination with its mirror image. But last year his team found another nonperiodic tile, known as the spectre, which is strictly chiral — that is, not only will it tile the plane without its mirror image, but it must be used in that way.

“Everyone believes in the normal law, the experimenters because they imagine that it is a mathematical theorem, and the mathematicians because they think it is an experimental fact.” — Gabriel Lippmann, in a letter to Henri Poincaré

(Thanks, Tom.)

The product of the distances from N equally spaced points on a unit circle to a point M that lies on the x axis at distance x from circle origin O and on a line passing through the first point, A₀, on the circle equals 1 – x^N.

Discovered by Roger Cotes (1682-1716). Here’s a proof.

Absolutely hands down one of the most beautiful places to see from space is the Caribbean. You see an entire rainbow of blue. From the light emerald green to the green-blue to the blue-green to the aquamarine to the slowly increasingly darker shades of blue down to the really deep colors that come with the depths of a really deep ocean. You can see all that at one time from space. It’s very curvy, it’s not harsh geometric lines. It’s swirls and whirls and all kinds of wavy lines. It looks like a piece of modern art.

— Astronaut Sandra Magnus, quoted in Ariel Waldman, What’s It Like in Space?, 2016

Swedish botanist Elias Tillander (1640–1693) was so “harassed by Neptune” during a trip across the Gulf of Bothnia from Stockholm to Turku that he made the return journey overland and changed his name to Til-Landz (“to land”).

Linnaeus named the evergreen plant Tillandsia after him — it cannot tolerate a damp climate.

(From Wilfrid Blunt, Linnaeus: The Compleat Naturalist, 2001.)

From Lee Sallows:

(Thanks, Lee!)

From reader Chris Smith:

Pick a three-digit number in which all the digits are different. Example: 314.

Now list every possible combination of two digits from the chosen number. In our example, these are 13, 14, 31, 34, 41, and 43.

Divide the sum of these two-digit numbers by the sum of the three digits in the original number, and you’ll always get 22. In our example, (13 + 14 + 31 + 34 + 41 + 43) / (3 + 1 + 4) = 176/8 = 22.

This works because 10a + b, 10a + c, 10b + a, 10b + c, 10c + a, and 10c + b sum to 22a + 22b + 22c = 22(a + b + c), so dividing by a + b + c will always give 22.

(Thanks, Chris.)

06/08/2024 Reader Tom Race points out that essentially the same trick can be performed using the entire number: If you add all six permutations of the original 3 digits, then divide that total by the sum of the 3 digits, the answer is always 222.

For example, using 561:

561 + 516 + 156 + 165 + 651 + 615 = 2664

5 + 6 + 1 = 12

2664 / 12 = 222

“This works because in the first sum each of the three digits (a, b and c) occurs twice in each of the three columns, so the sum is 222a + 222b + 222c = 222(a + b + c).” (Thanks, Tom.)