Stormy Weather

http://commons.wikimedia.org/wiki/File:5x5_Magic_Square_with_zero_water_retention.png
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:

http://en.wikipedia.org/wiki/File:Magic_Square_Graphic.png
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.

http://commons.wikimedia.org/wiki/File:Albrecht_D%C3%BCrer_-_Melencolia_I_(detail).jpg
Image: Wikimedia Commons
| Science & Math

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.”

| Science & Math

Finding Yourself

http://commons.wikimedia.org/wiki/File:Mozart_magic_flute.jpg

Twinkle, twinkle, little star,
How I wonder what you are.
Up above the world so high,
Like a diamond in the sky.
Twinkle, twinkle, little star,
How I wonder what you are.

Choose any word in the first two lines, count its letters, and count forward that number of words. For example, if you choose STAR, which has four letters, you’d count ahead four words, beginning with HOW, to reach WHAT. Count the number of letters in that word and count ahead as before. Continue until you can’t go any further. You’ll always land on YOU in the last line.

| Language · Science & Math

Soulmates

In 1966, asked to describe the person least likely to develop atherosclerosis, Cambridge research fellow Alan N. Howard answered, “A hypotensive, bicycling, unemployed, hypo-beta-lipoproteinic, hyper-alpha-lipoproteinic, non-smoking, hypolipaemic, underweight, premenopausal female dwarf living in a crowded room on the island of Crete before 1925 and subsisting on a diet of uncoated cereals, safflower oil, and water.”

Oxford physician Alan Norton added that her male counterpart was an ectomorphic Bantu who worked as a London bus conductor, had spent the war in a Norwegian prison camp, never ate refined sugar, never drank coffee, always ate five or more small meals a day, and was taking large doses of estrogen to check the growth of his prostate cancer.

“All these phrases mark correlations established in the last few years in a field of medical research which, in volume at least, is unsurpassed,” noted Richard Mould in Mould’s Medical Anecdotes. “The conflict of evidence is unequalled as well.”

| Science & Math

Ho

If this sentence is true, then Santa Claus exists.

If that sentence is true, then it’s the case that Santa Claus exists. But wait — in making this observation, we seem to have confirmed the truth of the original sentence. And if that sentence is true, then Santa Claus exists! Where is the error?

(By Raymond Smullyan.)

| Science & Math

Round Numbers

Every now and again one comes across an astounding result that closely relates two foreign objects which seem to have nothing in common. Who would suspect, for example, that on the average, the number of ways of expressing a positive integer n as a sum of two integral squares, x2 + y2 = n, is π?

— Ross Honsberger, Mathematical Gems III, 1977

| Science & Math

Times Square

Prove that the product of four consecutive positive integers cannot be a perfect square.

Click for Answer
| Puzzles · Science & Math

Dinner Charge

A curious effect produced by lightning is described to us by Dr. Enfield, writing from Jefferson, Iowa, U.S. A house which he visited was struck by lightning so that much damage was done. After the occurrence, a pile of dinner plates, twelve in number, was found to have every other plate broken. It would seem as if the plates constituted a condenser under the intensely electrified condition of the atmosphere. The particulars are, however, so meagre that it is difficult to decide whether the phenomenon was electrical or merely mechanical.

— Nature, June 12, 1902

| Oddities · Science & Math

A Body at Rest

newton mausoleum

In an anonymous letter to the London Times in 1825, Thomas Steele of Magdalen College, Cambridge, proposed enshrining Isaac Newton’s residence in a stepped stone pyramid surmounted by a vast stone globe. The physicist himself had died more than a century earlier, in 1727, and lay in Westminster Abbey, but Steele felt that preserving his home would produce a monument “not unworthy of the nation and of his memory”:

When travelling through Italy, I was powerfully struck by the unique situation and singular appearance of the Primitive Chapel at Assisi, founded by St. Francis.

As you enter the porch of the great Franciscan church, you view before you this small cottage-like chapel, standing directly under the dome, and perfectly isolated.

Now, Sir, among the many splendid improvements which are making in the capital, would it not be a noble, and perhaps the most appropriate, national monument which could be erected, if an azure hemispherical dome, or what would be better, a portion of a sphere greater than a hemisphere, supported on a massive base, were to be reared, like that of Assisi, over the house and observatory of the writer of the Principia?

The house might be fitted up in such a manner as to contain a council-chamber and library for the Royal Society; and it is perhaps not unworthy of being remarked, that it is not more than about two hundred yards distant from the University Club House.

Protected, by the means which I have described, from the dilapidating influence of rains and winds, the venerable edifice in which Newton studied, or was inspired, — that ‘palace of the soul,’ might stand fast for ages, a British monument more sublime than the Pyramids, though remote antiquity and vastness be combined to create their interest.

Steele wasn’t an architect, and he left the details to others, but he was imagining something enormous: In a subsequent letter to the Mechanics’ Magazine he wrote that “the base of my design [appears] to coincide with the base of St. Paul’s (a sort of crude coincidence of course, in consequence of the angle at the transept), and that the highest point of my designed building should, at the same time, appear to coincide with a point of the great tower of the cathedral, about 200 feet high — the height of the building which I propose to have erected.”

The plan never went forward, but the magazine endorsed the idea: “We need scarcely add, that there is no description of embellishment which might not be with ease introduced into the structure, so as to render it as perpetual a monument to the taste as it would be to the national spirit and gratitude of the British people.”

| Death · Science & Math