Cambridge mathematician Hallard T. Croft once asked whether it was possible to have a finite set of points in the plane with the property that the perpendicular bisector of any pair of them passes through at least two other points in the set.

In 1972 Leroy M. Kelly of Michigan State University offered the elegant solution above, a square with an equilateral triangle erected outward on each side (it also works if the triangles are erected inward).

“Croft is a great problemist,” Kelley said later. “He keeps putting out lists of problems and he keeps including that one. He’s trying to get the mathematical community to get a better example — one with more points in it. … Eight is the smallest number; and whether it’s the largest number is another question.”

So far as I know Croft’s question is still unanswered.

In 1894 Sir Francis Galton devised this simple machine to demonstrate the central limit theorem: Beads inserted at the top drop through successive rows of pegs, bouncing left or right as they hit each peg and landing finally in a row of bins at the bottom. Though the path of any given bead can’t be predicted, in the aggregate they form a bell curve. Delighted with this, Galton wrote:

Order in Apparent Chaos: I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the Law of Frequency of Error. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.

From Recreational Mathematics Magazine, 1961, a magic square of cards:

Each row, column, and main diagonal contains an ace, king, queen, and jack and all four suits. There are numerous other subsquares and symmetrical subsets of squares that have the same property, including the center 2 × 2 square and the four corner squares.

(Recreational Mathematics Magazine 34:5, 24-29, via Pi Mu Epsilon Journal, “Unusual Magic Squares,” 6:2 [Spring 1975], 54-55.)

A triangle can be covered by three smaller copies of itself. A square requires four smaller copies. But in general four will do: Any bounded convex set in the plane can be covered with four smaller copies of itself (and in fact the fourth copy is needed only in the case of parallelograms, like the square).

Is this true in every dimension? In 1957 Swiss mathematician Hugo Hadwiger conjectured that every n-dimensional convex body can be covered by 2ⁿ smaller copies of itself. But this remains an unsolved problem.

(Interestingly, Russian mathematician Vladimir Boltyansky showed that this problem is equivalent to one of illumination: How many floodlights does it take to illuminate an opaque convex body from the exterior? The number of floodlights turns out to equal the number of smaller copies needed to cover the body.)

It’s easy to see that a plane can be tiled with squares or hexagons arranged in regular ranks, but in 1936 Heinz Voderberg showed that it can also be tiled in a spiral formation. Each tile in the figure above is the same nine-sided shape, but together they form two “arms” that bound one another. If both arms are extended infinitely, they’ll cover the whole plane.

In 1955 Michael Goldberg showed that spirals might be devised with any even number of arms, and in 2000 Daniel Stock and Brian Wichmann did the same for odd numbers, so it’s now possible to devise a shape that will tile the plane in a spiral with any specified number of arms.

(Daniel L. Stock and Brian A. Wichmann, “Odd Spiral Tilings,” Mathematics Magazine 73:5 [December 2000], 339-346.)

In the 19th century scientists were increasingly interested in comparing personality with brain anatomy, but they faced a problem: Lower-class brains could be acquired fairly easily from hospital morgues, but people with exceptional brains had the means to protect them from the dissecting knife after death.

The solution was the Society of Mutual Autopsy (Société d’autopsie mutuelle), founded in 1876 “for the purpose of furnishing to the investigations of medicists brains superior to those of the common people.” Anatomists bequeathed their brains to each other, and the results of each investigation were read out to the other members of the club. (An early forerunner was Georges Cuvier, whose brain was found to weigh 1830 grams and displayed a “truly prodigious number of convolutions.”)

Similar “brain clubs” sprang up in Munich, Paris, Stockholm, Philadelphia, Moscow, and Berlin before the practice began to die out around World War I. Until then, writes anthropologist Frances Larson in Severed, her 2014 history of severed heads, “Members could die happy in the knowledge that their own brain would become central to the utopian scientific project they had pursued so fervently in life.”