Coincidence

https://commons.wikimedia.org/wiki/File:Cylinder_geometry.svg

This just caught my eye in an old issue of the Mathematical Gazette, a note from P.G. Wood. Suppose we’re designing a cylinder that’s closed at both ends and must encompass a given volume. What relative dimensions should we give it in order to minimize its surface area?

A young student thought, well, if we slice the cylinder with a plane that passes through its axis, the plane’s intersection with the cylinder will form a rectangle. And if we spin that rectangle, it’ll sweep out the surface area of the cylinder. So really we’re just asking: Among all rectangles of the same area, which has the smallest perimeter? A square. So the cylinder’s height should equal its diameter.

It turns out that’s right, but the student had overlooked something. The fact that the volume of the cylinder is fixed doesn’t imply that the area of the rectangle is fixed. We don’t know that.

Wood wrote, “We seem to have arrived at the right answer by rather dubious means.”

(P.G. Wood, “73.5 Interesting Coincidences?”, Mathematical Gazette 73:463 [1989], 33-33.)

Insight

Advice in problem solving:

“You must always invert.” — Carl Gustav Jacob Jacobi

“Whenever you can, count.” — Francis Galton

“Each problem that I solved became a rule, which served afterwards to solve other problems.” — Descartes

“By studying the masters, not their pupils.” — Niels Henrik Abel

“Truth is the offspring of silence and meditation. I keep the subject constantly before me and wait ’til the first dawnings open slowly, by little and little, into a full and clear light.” — Isaac Newton

Paperwork

https://commons.wikimedia.org/wiki/Category:Mary_Delany

Twice widowed, English artist Mary Delany (1700-1788) took up a remarkable new career in her 70s: She created a series of detailed and botanically accurate portraits of plants, devising them from tissue paper and coloring them by hand:

With the plant specimen set before her she cut minute particles of coloured paper to represent the petals, stamens, calyx, leaves, veins, stalk and other parts of the plant, and, using lighter and darker paper to form the shading, she stuck them on a black background. By placing one piece of paper upon another she sometimes built up several layers and in a complete picture there might be hundreds of pieces to form one plant. It is thought she first dissected each plant so that she might examine it carefully for accurate portrayal …

She kept it up until she lost her eyesight at 88, filling 10 volumes with 985 of these “paper mosaiks.” Eventually they were bequeathed to the British Museum.

(Ruth Hayden, Mrs Delany: Her Life and Her Flowers, 1980.)

Words and Numbers

https://pixabay.com/illustrations/pi-circle-diameter-circumference-1338559/

Strategies to memorize π sometimes rely on devising a sentence with words of representative lengths. Isaac Asimov offered this one:

How I want a drink, alcoholic, of course, after the heavy lectures involving quantum mechanics!

Count the letters in each word and you get 3.14159265358979. That’s not the best strategy, though: What shall we do about zero? And it’s hard to reel off the digits impressively for friends when you have to stop and count the letters in each word.

Arthur Benjamin, a mathematician at Harvey Mudd College, suggests using a phonetic code instead:

1 = t or th or d
2 = n
3 = m
4 = r
5 = l
6 = sh, ch, or j
7 = k or hard g
8 = f or v
9 = p or b
0 = s or z

Once we’ve memorized this list, we can turn a whole string of digits into a single word simply by inserting vowels between the consonants. So, for example, 314 could be represented by meter, motor, meteor, matter, mother, etc. The consonants h, w, and y, which don’t appear in the list, can be used like additional vowels.

Using this method, it takes only three sentences to encode the first 60 digits of π:

3. 1415 926 5 3 58 97 9 3 2 384 6264
My turtle Pancho will, my love, pick up my new mover Ginger.

3 38 327 950 2 8841 971
My movie monkey plays in a favorite bucket.

69 3 99 375 1 05820 97494
Ship my puppy Michael to Sullivan’s backrubber!

Now we just need a way to remember the sentences …

(Arthur T. Benjamin, “A Better Way to Memorize Pi: The Phonetic Code,” Math Horizons 7:3 [February 2000], 17.)

Revelation

The invention of logarithms came on the world as a bolt from the blue. No previous work had led up to it, foreshadowed it, or heralded its arrival. It stands isolated, breaking in upon human thought abruptly without borrowing from the work of other intellects or following known lines of mathematical thought.

— Lord Moulton on the 300th anniversary of John Napier’s 1614 book Description of the Wonderful Canon of Logarithms. E.W. Hobson called the invention “one of the very greatest scientific discoveries that the world has seen.”

Ascendant

https://commons.wikimedia.org/wiki/File:Usain_Bolt_2011-09-04_001.jpg
Image: Wikimedia Commons

Usain Bolt is such a great sprinter that his distinctions may extend to other worlds.

In 2013, University of Leicester physics undergraduate Hannah Lerman and her colleagues determined that the Jamaican athlete was one of the few humans who could get aloft on Saturn’s moon Titan with wings strapped to his arms.

Factoring in Titan’s gravity and atmospheric density, Lerman found that a person could take flight in a normal-sized wingsuit only if they could run at 11 meters per second.

“This speed has been reached but only by the fastest human runners, for example, Usain Bolt, who ran almost 12 m/s,” Lerman wrote. “For an average human to take off with the standard wingsuit they would require some sort of propulsion device to give them enough speed to take off.”

(H. Lerman, B. Irwin, and P. Hicks, “P5_1 You Can Fly,” Journal of Physics Special Topics, University of Leicester, Oct. 22, 2013.)

Löb’s Paradox

A paradox by the German mathematician Martin Löb:

Let A be any sentence. Let B be the sentence: ‘If this sentence is true, then A.’ Then a contradiction arises.

Here’s the contradiction. B makes the assertion “If B is true, then A.” Now consider this argument. Assume B is true. Then, by B, since B is true, A is true. This argument shows that, if B is true, then A. But that’s exactly what B had asserted! So B is true. And therefore, by B, since B is true, A is true. And thus every sentence is true, which is impossible.

(Lan Wen, “Semantic Paradoxes as Equations,” Mathematical Intelligencer 23:1 [December 2001], 43-48.)

Proof Without Words

schaer proof

In the January 2001 issue of the College Mathematics Journal, University of Calgary mathematician Jonathan Schaer offers this simple proof that arctan 1 + arctan 2 + arctan 3 = π. The sum of the angles of this large triangle is 180°. And the diagram shows that its lower left angle is arctan 3, its lower right angle is arctan 2, and its top angle is part of an isosceles right triangle. So arctan 1 + arctan 2 + arctan 3 = 180°, or π radians. “No words, even no symbols!”

(“Miscellanea,” College Mathematics Journal 32:1, 68-71.)

Early Times

Welsh mathematician Robert Recorde’s 1543 textbook Arithmetic: or, The Ground of Arts contains a nifty algorithm for multiplying two digits, a and b, each of which is in the range 5 to 9. First find (10 – a) × (10 – b), and then add to it 10 times the last digit of a + b. For example, 6 × 8 is (4 × 2) + (10 × 4) = 48.

This works because (10 – a)(10 – b) + 10(a + b) = 100 + ab, and it saves the student from having to learn the scary outer reaches of the multiplication table — they only have to know how to multiply digits up to 5.

(From Stanford’s Vaughan Pratt, in Ed Barbeau’s column “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 38:1 [January 2007], 43-46.)

Skyfall

https://pixabay.com/photos/moon-full-moon-sky-night-sky-lunar-1859616/

Iowa State University mathematician Alexander Abian was a quiet man with a bold idea: He believed that blowing up the moon would solve most of humanity’s problems. In thousands of posts on Usenet, he maintained that destroying the moon would eliminate Earth’s wobble, canceling the seasons and associated calamities such as hurricanes and snowstorms.

“You make a big hole by deep drilling, and you put there atomic explosive,” he explained in 1991. “And you detonate it — by remote control from Earth.”

“I was questioned about it,” wrote English astronomer Patrick Moore in Fireside Astronomy (1993). “I pointed out, gently, that even if the Moon were removed it would not alter the tilt of the Earth’s axis in the way that the professor seems to believe. Moreover, the energy needed to destroy the Moon would certainly destroy the Earth as well, even if we had the faintest idea of how to do it. The British Meteorological Office commented that a moonless Earth would be ‘bleak and tideless’, and a spokesman for the British Association for the Advancement of Science, struggling nobly to keep a straight face, asked what would happen if the experiment went wrong. Predictably, Professor Abian was unrepentant. ‘People don’t seem prepared to sacrifice the Moon for a better climate. It is inevitable that the genius of man will one day accept my ideas.'”

For better or worse, he maintained this position until his death in 1999. “I am raising the petulant finger of defiance to the solar organization for the first time in 5 billion years,” he said. “Those critics who say ‘Dismiss Abian’s ideas’ are very close to those who dismissed Galileo.”