Science & Math


A visitor to Niels Bohr’s cottage noticed a horseshoe nailed over the door.

“Surely you don’t expect that a horseshoe will bring good luck?” asked the visitor.

“No, I don’t,” Bohr said. “But they say it works even if you don’t believe in it.”

See The Misfortune Field.

The Look and Say Sequence

What’s the key to this curious sequence of numbers?

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …

When read aloud, each term describes the one that precedes it. The first term consists of “one 1,” the second of “two 1s,” the third of “one 2, then one 1,” and so on.

That seems pretty arbitrary, but it opens a door into an unsuspected mathematical universe. Start with any number (except 22, an obvious dead end) and it will produce a string of digits that lengthens by about 30 percent with each generation — indeed, the percentage approaches a predictable constant (30.3577269 …) as the length approaches infinity.

More amazingly, the growing string will organize itself into a series of recognizable finite substrings that evolve predictably with each generation. John Horton Conway, who discovered all this, identified 92 such substrings, which he named after the chemical elements. Thus “uranium” (3) decays into “protactinium” (13), which becomes “thorium” (1113), and so on.

Thus an infinitely complex universe can arise from simply reading the number 1 aloud.

The Euler Characteristic

For any convex polyhedron,

vertices – edges + faces = 2

In 1988, readers of the Mathematical Intelligencer judged this the world’s second most beautiful theorem — behind Euler’s identity.


  • Georgia, Massachusetts, and Connecticut didn’t ratify the Bill of Rights until 1939.
  • Wilt Chamberlain never fouled out of a game.
  • 3864 = 3 × (-8 + 64)
  • What’s the opposite of “not in”?
  • Alaska has a longer coastline than all other U.S. states combined.
  • “To do nothing is also a good remedy.” — Hippocrates

Same Difference

math notes - equivalent operations

A Better Nature

Trisect the angles of any triangle and you’ll find an equilateral triangle at its heart.

This theorem has a curious cousin: If you trisect the sides of any triangle and erect an equilateral triangle outwardly on the middle third of each leg, then the outermost vertices of these equilateral triangles will themselves form an equilateral triangle.

The Thatcher Effect

thatcher effect

When we look at another person’s face, her eyes and mouth convey the most information about her mood.

Indeed, when a face is inverted we can have trouble recognizing it because we can’t read its expression.

So in 1980 University of York psychologist Peter Thompson tried inverting everything but the eyes and mouth.

Most people can recognize the face at left and assign a mood to it, but they’re often surprised to see it right side up.

“Further research into this illusion might help determine whether face recognition is a serial or parallel process,” Thompson wrote in Perception that summer. “It might even tell us something about Margaret Thatcher.”

Scholars and Sense

O Cleinias, are those who learn the wise or the ignorant?

He answered that those who learned were the wise.

Euthydemus proceeded: There are some whom you would call teachers, are there not?

The boy assented.

And they are the teachers of those who learn — the grammar-master and the lyre master used to teach you and other boys; and you were the learners?


And when you were learners you did not as yet know the things which you were learning?

No, he said.

And were you wise then?

No, indeed, he said.

But if you were not wise you were unlearned?


You then, learning what you did not know, were unlearned when you were learning?

The youth nodded assent.

Then the unlearned learn, and not the wise, Cleinias, as you imagine.

— Plato, Euthydemus

Math Notes

multigrade equation

Never Mind

“It has been asserted (by C.S. Lewis, for instance) that no determinist rationally can believe in determinism, for if determinism is true, his beliefs were caused, including his belief in determinism. The idea seems to be that the causes of belief, perhaps chemical happenings in the brain, might be unconnected with any reasons for thinking determinism true. They might be, but they need not be. The causes might ‘go through’ reasons and be effective only to the extent that they are good reasons.”

— Robert Nozick, “Reflections on Newcomb’s Paradox,” 1974

“If … [determinism] is true, then the intellectual or cognitive operations of its upholders, including their choice or decision to maintain the thesis, … are themselves only the effects of inexorable forces. But if this is so, why should the thesis … be accepted as valid or true?”

— Alan Gewirth, Reason and Morality, 1978

Pandigital Approximations

The digits 1-9 can work some impressive tricks:
The first formula, found by B. Ziv in 2004, produces the first 10 digits of pi.

The second, astonishingly, reproduces e to 18,457,734,525,360,901,453,873,570 decimal places. It was discovered by Richard Sabey, also in 2004.

(Thanks, Robin.)

Party of One

“A man wrote to say that he accepted nothing but Solipsism, and added that he had often wondered it was not a more common philosophy. Now Solipsism simply means that a man believes in his own existence, but not in anybody or anything else. And it never struck this simple sophist, that if his philosophy was true, there obviously were no other philosophers to profess it.”

— G.K. Chesterton, St. Thomas Aquinas, 1933

Digit Acrobatics

214358976 = (3 + 6)2 + (4 + 7)8 + (5 + 9)1

“Triangle Rhyme”

Although the altitudes are three,
Remarks my daughter Rachel,
One point’ll lie on all of them:
The orthocenter H’ll.

By mathematician Dwight Paine of Messiah College, 1983.

(Further recalcitrant rhymes: month, orange. W.S. Gilbert weighs in.)


  • River Phoenix was born River Bottom.
  • Every natural number is the sum of four squares.
  • What happens if Pinocchio says, “My nose will grow now”?
  • Shakespeare has no living descendants.
  • “All generalizations are dangerous — even this one.” — Dumas

Math Notes

000002569 is prime.

You Are Here

In “Partial Magic in the Quixote,” Borges writes:

Let us imagine that a portion of the soil of England has been levelled off perfectly and that on it a cartographer traces a map of England. The job is perfect; there is no detail of the soil of England, no matter how minute, that is not registered on the map; everything has there its correspondence. This map, in such a case, should contain a map of the map, which should contain a map of the map of the map, and so on to infinity.

This sequence tends to a single point, the point on the map that corresponds directly to the point it represents in the territory.

Cover England entirely with a 1:1 map of itself, then crumple the map into a ball. So long as it remains in England, the balled map will always contain at least one point that lies directly above the corresponding point in England.

See Garganta and Papered Over.



Draw a semicircle and surmount it with two smaller semicircles.

A line drawn through A, at any angle, will divide the perimeter precisely in half.

This probably has some romantic symbolism, but I’m not very good at that stuff.

Squares Into Squares

squares into squares


Imagine you have a little robot that holds a pencil. Set it down on a sheet of paper and give it these instructions:

  1. Move forward 3 units and turn right.
  2. Move forward 1 unit and turn right.
  3. Move forward 2 units and turn left.
  4. Move forward 1 unit and turn left.
  5. Move forward 2 units and turn right.
  6. Repeat.

If the robot makes its turns at 90° angles, it will produce this figure:


But, remarkably, if it turns at 120° it will draw this:

star of david

The Grid Flower

ellipses and hyperbolas

Any pair of points define an infinity of ellipses and an infinity of hyperbolas.

The ellipses do not touch one another, nor do the hyperbolas.

But every ellipse meets every hyperbola at a right angle.

The Black Hole

unilluminable room

If Satan plays miniature golf, this is his favorite hole. A ball struck at A, in any direction, will never find the hole at B — even if it bounces forever.

The idea arose in the 1950s, when Ernst Straus wondered whether a room lined with mirrors would always be illuminated completely by a single match.

Straus’ question went unanswered until 1995, when George Tokarsky found a 26-sided room with a “dark” spot; two years later D. Castro offered the 24-sided improvement above. If a candle is placed at A, and you’re standing at B, you won’t see its reflection anywhere around you — even though you’re surrounded by mirrors.

The Franklin Square

franklin magic square

In a 1769 letter, Ben Franklin describes a magic square he devised in his youth. The magic total of 260 can be reached by adding the numbers in each row or column, as in a normal magic square. But “bent rows” (shaded) produce the same total, even when “wrapped across” the border of the table. This works in all four directions.

Further: Half of each row or column sums to half of 260, as does any 2×2 subsquare. And the four corners and the four center squares sum to 260. (Alas, the main diagonals don’t, so this doesn’t strictly qualify as a magic square by the modern definition.)

Interestingly, no one knows how Franklin created the square. Many methods have been devised, but none apparently as quick as his, which he claimed could generate them “as fast as he could write.”

The Perfect Family

Take any Platonic solid, join the centers of its faces, and, charmingly, you get another Platonic solid. The cube and the octahedron produce one another, as do the dodecahedron and the icosahedron, and the tetrahedron produces another tetrahedron.

Bonus factoid: If you inscribe a dodecahedron and an icosahedron in the same sphere, the dodecahedron will occupy more of the sphere’s volume. It has fewer faces than the icosahedron, but its faces are more nearly circular, so it fits the sphere more snugly.

See The Pup Tent Problem.