# A Curious Equality

The area of the largest equilateral triangle that fits inside a unit square exactly equals the side of the largest square that fits inside a unit equilateral triangle.

This proof is due to John Conway. The largest square that fits inside a unit equilateral triangle is shown at the top; the side of the square is $2\sqrt{3}-3$. The largest equilateral triangle that fits inside a unit square is one that’s canted at a 15° angle, shown at the bottom.

Now, the area of the shaded parallelogram is just its base times its height, or $2\sqrt{3}-3$. And the parallelogram can be dissected into three pieces that fit precisely into the canted triangle. So the side of the square at the top equals the area of the triangle at the bottom.

Do animals reason? British psychologist C. Lloyd Morgan offered the case of his own dog:

Tony, the fox-terrier, already introduced to my readers, when he wants to go out into the road, puts his head under the latch of the gate, lifts it, and waits for the gate to swing open. Now an observer of the dog’s intelligent action might well suppose that he clearly perceived how the end in view was to be gained, and the most appropriate means for effecting his purpose. The following chain of ideas might be supposed to pass through the dog’s mind, not, indeed, in a clear-cut logical form, but at any rate in a rough and practically serviceable way: ‘Why does that gate remain shut? The latch holds it. I’ll lift the latch. Now it is no longer held, therefore it swings open.’ But is it necessary to assume that there were ideas involving, even in the most rudimentary way, the why and the wherefore? May not the action be quite well explained on the hypothesis that the dog acted under the sole guidance of sense-experience?

Two observers might debate that question for some time, Morgan wrote, one arguing that Tony’s feat must be based on experience and the other insisting that it showed an understanding of how the gate operates. But Morgan himself had had the benefit of seeing the trick evolve. “I was sitting at a window above the garden, and heard the dog put out of the door. I therefore watched him. He ran up and down the low wall, and put his head out between the iron bars, now here, now there, now elsewhere, anxiously gazing into the road. This he did for quite three or four minutes. At length it so happened that he put out his head beneath the latch, which, as I have said, is at a convenient height for his doing so, being about a foot above the level of the wall. The latch was thus lifted. He withdrew his head, and began to look out elsewhere, when he found that the gate was swinging open, and out he bolted.”

So Tony’s trick was the product of a fortunate accident, not abstract reasoning. Morgan proposed a general rule: “In no case is an animal activity to be interpreted in terms of higher psychological processes if it can be fairly interpreted in terms of processes which stand lower in the scale of psychological evolution and development.” This has become known as Morgan’s canon.

(From Morgan’s 1903 book An Introduction to Comparative Psychology.)

# Dizzy

Russian science writer Yakov Perelman asks: How many times must an 8-toothed cogwheel rotate on its axis to circle around a 24-toothed cogwheel?

# Bright and Early

Meteors are more commonly seen between midnight and dawn than between dusk and midnight. Why?

# An Odd Fractal

In 1653, Blaise Pascal composed a triangular array in which the number in each cell is the sum of the two directly above it:

In 1915, Polish mathematician Waclaw Sierpinski described an equilateral triangle in which the central fourth is removed and the same procedure is applied to all the succeeding smaller triangles. Perplexingly, the resulting structure has zero area:

Interestingly, if the odd numbers in Pascal’s triangle are shaded, they produce an approximation to Sierpinski’s triangle:

And as this triangle grows toward infinity, it becomes Sierpinski’s triangle — an arrangement of numbers that takes the shape of a geometrical object.

If two squares share a common vertex, then the centers of the squares and the midpoints of the linking segments shown form another square.

Discovered by mathematicians Paul Finsler and Hugo Hadwiger.

# Van Aubel’s Theorem

Draw any quadrilateral and build a square on each of its sides.

Connect the centers of opposite squares, and these two line segments will be perpendicular and of equal length.

# The Five Circles Theorem

If the centers of five circles lie along the circumference of a sixth so that they overlap like the links of a chain, and if each intersects its neighbor on the sixth circle as well, then drawing lines as shown through the remaining intersections will form a pentagram whose points lie on the five circles.

A 19th-century schoolchild defined a circle as “a round straight line with a hole in the middle.”

# Catalog

Entomologist Alexandre Girault expressed himself in his work. Of the 500 genera he named, many honored artists, poets, and writers whom he admired: Davincia, Shakespearia, Beethovena, Mozartella, Emersonia, Raphaelana, Ovidia, Goetheana, Thoreauella, Tennysoniana, Bachiana, Keatsia, Plutarchia, Schilleria, Aeschylia, Aligheria, Thalesanna, Rubensteina, Carlyleia, Herodotia, Cowperella, and Froudeana.

To mock his supervisor Johann Illingworth, he invented a parasitic mymarid wasp, Shillingsworthia shillingsworthi, which he described as an ephemeral creature lacking a head, abdomen, or mandibles and found only in “the chasms of Jupiter” — in other words, a nonentity. He called it “an airy species whose flight cannot be followed except by the winged mind.”

He understood even women through entomology — one of his privately printed works describes a new species of human, Homo perniciosus, known only from the female sex:

Abnormal female (loveless, without offspring); heart functionless; mammae aborted; psychology novel (as supposed) but artificial; gay, high-coloured, feral, brass-cheeked, shape lovely like Woman but nature hard (selfless, thoughtless, proud, unsympathetic, irresponsible, aggressive, irritant, insensible, luxurious, pugnacious, over-active, inquisitive, mischievous, voracious and even carnivorous; antagonistic, ungentle, immodest, critical, competitive, poisonous); conduct unstable (even inclined to treachery), the lips compressed, body strong. Everywhere but rare in natural habitat.

He was prickly, but he was dedicated — he published much of his work at his own expense, and many of his type specimens are retained today by the Queensland Museum. “Research is a labour of love,” he wrote. “Strange then to find it all done nowadays as a labour of wages! Must love, too, be a matter of cash?”

# Clifford’s Circle Theorems

Let four circles (blue) pass through a single point, M. Each pair of these circles intersect at a second point (pink). Each three of the four blue circles will have three pink points among them; these trios of pink points define four new circles (brown), which intersect in a single point, P.

If we start with five circles passing through a single point M, then we can apply the procedure above to each subset of four of them. This will produce five points P that all lie on a single circle.

If we start with six circles that all pass through a single point M, then each subset of five of them defines a new circle, as we’ve just seen. These six new circles all pass through a single point.

Remarkably, this pattern continues forever. It was discovered by the English geometer William Kingdon Clifford.