Brothers in Binary

A number is said to be perfect if it equals the sum of its divisors: 6 is divisible by 1, 2, and 3, and 1 + 2 + 3 = 6.

St. Augustine wrote, “Six is a number perfect in itself, and not because God created all things in six days; rather the converse is true; God created all things in six days because this number is perfect, and it would have been perfect even if the work of the six days did not exist.”

Perfect numbers are rare. No one knows whether an infinite quantity exist, and no one knows whether any of them are odd. The early Greeks knew the first four, and in the ensuing two millennia we’ve uncovered only 44 more. But they have one thing in common — they reveal a curious harmony when expressed in base 2:

brothers in binary


“We have not the reverent feeling for the rainbow that a savage has, because we know how it is made. We have lost as much as we gained by prying into that matter.” — Mark Twain

“At last I fell fast asleep on the grass & awoke with a chorus of birds singing around me, & squirrels running up the trees & some Woodpeckers laughing, & it was as pleasant a rural scene as ever I saw, & I did not care one penny how any of the beasts or birds had been formed.” — Charles Darwin, letter to his wife, April 28, 1858


Launched in 1976, NASA’s Laser Geodynamic Satellite needed a stable orbit to permit precise measurements of continental drift, so its designers gave it a high trajectory and a heart of solid brass. As a result, it’s not expected to return to Earth for 8 million years. That raised an interesting challenge: What message could we attach to the satellite that might be intelligible to our descendants or successors, who might recover it thousands of millennia in the future?

Tasked with that problem, Carl Sagan came up with the “greeting card” at left, which is affixed to LAGEOS on a small metal plaque. Using it, whoever comes upon the plaque can calculate roughly the time between his own epoch and ours. In Sagan’s words, the card says, “A few hundred million years ago the continents were all together, as in the top drawing. At the time LAGEOS was launched the map of the Earth looks as in the middle drawing. Eight million years from now, when LAGEOS should return to Earth, we figure the continents will appear as in the bottom drawing. Yours truly.”


rectangle theorem

For any rectangle, the sum of the squares of the distances from any point P to two opposite corners is equal to the sum of the squares of the distances from that point to the two other corners (so, above, a2 + c2 = b2 + d2). This remains true whether the point is inside or outside the rectangle, on a side or a corner, or even outside the plane.

Pushkin wrote, “Inspiration is needed in geometry, just as much as in poetry.”

Correlation, Causation

mould storks

From Richard F. Mould’s Introductory Medical Statistics — this graph plots the population of Oldenburg, Germany, at the end of each year 1930-1936 against the number of storks observed in that year.

Does this explain the storks’ presence? Not necessarily: In 1888 J.J. Sprenger noted, “In Oldenburg there is a curious theory that the autumnal gatherings of the storks are in reality Freemasons’ meetings.”

Digit Work

A useful system of finger reckoning from the Middle Ages:

To multiply 6 x 9, hold up one finger, to represent the difference between the 5 fingers on that hand and the first number to be multiplied, 6.

On the other hand, hold up four fingers, the difference between 5 and 9.

Now add the number of extended fingers on each hand to get the first digit of the answer (1 + 4 = 5), and multiply the number of closed fingers on each hand to get the second (4 × 1 = 4). This gives the answer, 54.

In this way one can multiply numbers between 6 and 9 while knowing the multiplication table only up to 5 × 5.

A similar system could be used to multiply numbers between 10 and 15. To multiply 14 by 12, extend 4 fingers on one hand and 2 on the other. Add them to get 6; add 10 times that sum to 100, giving 160; and then add the product of the extended fingers, 4 × 2, to get 168.

This system reflects the fact that xy = 10 [(x – 10) + (y – 10)] + 100 + (x – 10)(y – 10).

(From J.T. Rogers, The Story of Mathematics, 1968.)

Stormy Weather
Image: Wikimedia Commons

Take an ordinary magic square and imagine that the number in each cell denotes its altitude above some common underlying plane. And now suppose that it begins to rain, with an equal amount of water falling onto each cell. What happens? In the square at left, the water cascades from square 25 down to square 21, and thence down to 10, 7, 2, and into space; because there are no “lowlands” on this landscape, no water is retained. (Water flows orthogonally, not diagonally, and it pours freely over the edges of the square.)

By contrast, in the square on the right a “pond” forms that contains 69 cubic units of water — as it happens, the largest possible pond on a 5×5 square.

With the aid of computers, these imaginary landscapes can be “terraformed” into surprisingly detailed shapes. Craig Knecht, who proposed this area of study in 2007, created this 25×25 square in 2012:
Image: Wikimedia Commons

Next year will mark the 500th anniversary of the famously fertile magic square in Albrecht Dürer’s 1514 engraving Melancholia — a fact that Knecht has commemorated in the shape of the ponds on the 14×14 square at right.
Image: Wikimedia Commons

Partial Credit

On his 36th birthday, feeling that his most fertile years were behind him, mathematician Abram Besicovitch said, “I have had four-fifths of my life.”

At age 59 he was elected to the Rouse Ball Chair of Mathematics at Cambridge.

When J.C. Burkill reminded him of his earlier remark, he said, “Numerator was correct.”