In 1837 English journalist Albany Fonblanque wrote, “Sir Robert Peel was a smooth round peg, in a sharp-cornered square hole, and Lord Lyndenurst is a rectangular square-cut peg, in a smooth round hole.”

Which of these is the better fit? In other words, which is larger, the ratio of the area of a circle to a circumscribed square, or the area of a square to a circumscribed circle?

In two dimensions, these ratios work out to π/4 and 2/π, respectively, so a round peg fits better into a square hole than a square peg into a round hole.

But, strangely, Berkeley mathematician David Singmaster discovered in 1964 that this is true only in dimensions less than 9. For n ≥ 9 the n-dimensional unit cube fits more closely into the n-dimensional unit sphere than vice versa.

There’s a moral in there, but I don’t know what it is.

(David Singmaster, “On Round Pegs in Square Holes and Square Pegs in Round Holes,” Mathematics Magazine 37:5 [November 1964], 335-337.)

When a high voltage difference is applied, a liquid bridge of deionized water can sustain itself between two beakers even when they’re separated by 25 millimeters.

British engineer William Armstrong first reported this in an 1893 lecture. The phenomenon is known to be founded in surface polarization, but it’s still not completely understood.

Postscript of a letter from Benjamin Franklin to the Abbé André Morellet, July 1779:

P.S. To confirm still more your piety and gratitude to Divine Providence, reflect upon the situation which it has given to the elbow. You see in animals, who are intended to drink the waters that flow upon the earth, that if they have long legs, they have also a long neck, so that they can get at their drink without kneeling down. But man, who was destined to drink wine, is framed in a manner that he may raise the glass to his mouth. If the elbow had been placed nearer the hand, the part in advance would have been too short to bring the glass up to the mouth; and if it had been nearer the shoulder, that part would have been so long that when it attempted to carry the wine to the mouth it would have overshot the mark, and gone beyond the head; thus, either way, we should have been in the case of Tantalus. But from the actual situation of the elbow, we are enabled to drink at our ease, the glass going directly to the mouth.

“Let us, then, with glass in hand, adore this benevolent wisdom; — let us adore and drink!”

This just caught my eye in an old issue of the Mathematical Gazette, a note from P.G. Wood. Suppose we’re designing a cylinder that’s closed at both ends and must encompass a given volume. What relative dimensions should we give it in order to minimize its surface area?

A young student thought, well, if we slice the cylinder with a plane that passes through its axis, the plane’s intersection with the cylinder will form a rectangle. And if we spin that rectangle, it’ll sweep out the surface area of the cylinder. So really we’re just asking: Among all rectangles of the same area, which has the smallest perimeter? A square. So the cylinder’s height should equal its diameter.

It turns out that’s right, but the student had overlooked something. The fact that the volume of the cylinder is fixed doesn’t imply that the area of the rectangle is fixed. We don’t know that.

Wood wrote, “We seem to have arrived at the right answer by rather dubious means.”

(P.G. Wood, “73.5 Interesting Coincidences?”, Mathematical Gazette 73:463 [1989], 33-33.)