Piece Work
Tangrams can demonstrate the Pythagorean theorem. The yellow figure in the diagram above is a right triangle; the seven pieces that make up the square on the hypotenuse can be rearranged to form squares on the other two sides.
The third-century mathematician Liu Hui used to explain the theorem by dissecting and rearranging squares. Proper tangrams did not appear until centuries later, but modern Chinese mathematician Liu Dun writes, “We can hypothesize that the inventor of the Tangram, if not a mathematician, was at least inspired or enlightened by” this practice.
(From Jerry Slocum, The Tangram Book, 2003.)
The Balmis Expedition
Image: Wikimedia Commons
Smallpox ravaged the New World for centuries after the Spanish conquest. In 1797 Edward Jenner showed that exposure to the cowpox virus could protect one against the disease, but the problem remained how to transport cowpox across the sea. In 1802 Charles IV of Spain announced a bold plan — 22 orphaned children would be sent by ship; after the first child was inoculated, his skin would exude fluid that could be passed to the next child. By passing the live virus from arm to arm, the children formed a transmission chain that could transport the vaccine in an era before refrigeration and other modern technology was available.
It worked. Over the next 10 years Spain spread the vaccine throughout the New World and to the Philippines, Macao, and China. Oklahoma State University historian Michael M. Smith writes, “These twenty-two innocents formed the most vital element of the most ambitious medical enterprise any government had ever undertaken.” Jenner himself wrote, “I don’t imagine the annals of history furnish an example of philanthropy so noble, so extensive as this.”
Likewise
A charming little scene from mathematical history — in 1615 Gresham College geometry professor Henry Briggs rode the 300 miles from London to Edinburgh to meet John Napier, the discoverer of logarithms. A contemporary witnessed their meeting:
He brings Mr. Briggs up into My Lord’s chamber, where almost one quarter of an hour was spent, each beholding the other with admiration, before one word was spoke: at last Mr. Briggs began. ‘My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto Astronomy, viz. the Logarithms: but my Lord, being by you found out, I wonder nobody else found it before, when now being known it appears so easy.’
Their friendship was fast but short-lived: The first tables were published in 1614, and Napier died in 1617, perhaps due to overwork. In his last writings he notes that “owing to our bodily weakness we leave the actual computation of the new canon to others skilled in this kind of work, more particularly to that very learned scholar, my dear friend, Henry Briggs, public Professor of Geometry in London.”
A Magic Jigsaw
Image: Wikimedia Commons
A “self-interlocking” geomagic square by Lee Sallows. The 16 lettered pieces pave a single large square, and smaller squares can be produced by various groups of four pieces — those drawn from each row, column, and long diagonal, and 10 other symmetrically chosen quartets.
“A Rough Justice”
British inventor Sir Robert Watson-Watt pioneered the development of radar, a contribution that helped the Royal Air Force win the Battle of Britain. Ironically, after the war he was pulled over for speeding by a Canadian policeman wielding a radar gun. His wife tried to point out the absurdity of the situation, but the officer wasn’t interested, and the couple drove away with a $12.50 fine. Watson-Watt wrote this poem:
Pity Sir Robert Watson-Watt,
strange target of this radar plot
And thus, with others I can mention,
the victim of his own invention.
His magical all-seeing eye
enabled cloud-bound planes to fly
but now by some ironic twist
it spots the speeding motorist
and bites, no doubt with legal wit,
the hand that once created it.
Oh Frankenstein who lost control
of monsters man created whole,
with fondest sympathy regard
one more hoist with his petard.
As for you courageous boffins
who may be nailing up your coffins,
particularly those whose mission
deals in the realm of nuclear fission,
pause and contemplate fate’s counter plot
and learn with us what’s Watson-Watt.
(Thanks, Chris.)
Augury
Ohio State University philosopher Stewart Shapiro relates a puzzling experience that a friend once encountered in a physics lab. “The class was looking at an oscilloscope and a funny shape kept forming at the end of the screen. Although it had nothing to do with the lesson that day, my friend asked for an explanation. The lab instructor wrote something on the board (probably a differential equation) and said that the funny shape occurs because a function solving the equation has a zero at a particular value. My friend told me that he became even more puzzled that the occurrence of a zero in a function should count as an explanation of a physical event, but he did not feel up to pursuing the issue further at the time.
“This example indicates that much of the theoretical and practical work in a science consists of constructing or discovering mathematical models of physical phenomena. Many scientific and engineering problems are tasks of finding a differential equation, a formula, or a function associated with a class of phenomena. A scientific ‘explanation’ of a physical event often amounts to no more than a mathematical description of it, but what on earth can that mean? What is a mathematical description of a physical event?”
What right do we have to presume that the natural world will hew to mathematical laws? And why does the universe oblige us so graciously by doing so? Repeatedly, mathematicians have developed abstract structures and concepts that have later found unexpected applications in science. How can this happen?
“It is positively spooky how the physicist finds the mathematician has been there before him or her.” — Steven Weinberg
“I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with the original thing.” — Richard Feynman
“One cannot escape the feeling that these mathematical formulae have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.” — Heinrich Hertz
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” — Eugene Wigner
(From Stewart Shapiro, Thinking About Mathematics, 2000; also his paper “Mathematics and Reality” in Philosophy of Science 50:4 [December 1983].)
Misc
Image: Wikimedia Commons
- Denver International Airport is larger than Manhattan.
- C.S. Lewis, Aldous Huxley, and John F. Kennedy died on the same day.
- Shakespeare mentions America only once, in Act 3, Scene 2 of The Comedy of Errors.
- π4 + π5 ≈ e6
- “All styles are good except the boring kind.” — Voltaire
(Thanks, Larry.)
Versatile
What’s unusual about this magic square?
It works just as well upside down:
From Royal V. Heath, Scripta Mathematica, March-June 1951. See Topsy Turvy.
Small Business
Image: Wikimedia Commons
To help interest young students in chemistry, James Tour of Rice University devised “NanoPutians,” organic molecules that take the form of stick figures. The body is a series of carbon atoms that join two benzene rings; the arms and legs are acetylene units, each terminating in an alkyl group; and the head is a 1,3-dioxolane ring.
This gets even better — by using microwave irradiation, Tour found a way to vary the heads, creating a range of NanoProfessionals:
Image: Wikimedia Commons
The synthesis is detailed on the Wikipedia page.
Lights Out
I pass on to Eclipses. When the Moon (see above) gets between the Earth (see below) and the Sun (do what you like), the resulting phenomenon is called an Eclipse of the Sun. When the Sun gets between the Earth and the Moon there will be the devil to pay. It will be called the Eclipse of the Earth and is likely to be total.
— H.F. Ellis, So This Is Science!, 1932
Equal Opportunity
Can two dice be weighted so that the probability of each of the numbers 2, 3, …, 12 is the same?
Cause for Alarm
Sherlock Holmes is walking through the valley of Reichenbach Fall. On a clifftop overhead, Moriarty has perched a boulder. When he pushes it, it will have a 90 percent chance of killing Holmes. Just as he is about to send it over the edge, Watson arrives at the clifftop. Watson can’t see Holmes, so he’s not able to push the boulder safely clear, but he reasons that it’s better to push the boulder in a random direction than to let Moriarty aim it carefully. So he pushes the boulder off the cliff in such a way that Holmes’ chance of dying is reduced to 10 percent.
Unfortunately the boulder crushes Holmes anyway. Watson’s push decreased the chance of Holmes’ death, but it also caused it.
What are we to make of this? Generally speaking, it seems true to say that Pre-emptive pushing prevents death by crushing. That is, Watson’s push was of the sort that made it less likely that Holmes would die — if the scenario were re-enacted many times, with the boulder pushed sometimes by Watson, sometimes by Moriarty, Watson-type pushes would result in fewer deaths. But it also seems true to say that Watson’s pushing the rock caused Holmes to die. But cause and prevent are antonyms. How can both of these statements be true?
(Christopher Read Hitchcock, “The Mishap at Reichenbach Fall: Singular vs. General Causation,” Philosophical Studies, June 1995.)
Half and Half
A bisecting arc is one that bisects the area of a given region. “What is the shortest bisecting arc of a circle?” Murray Klamkin asked D.J. Newman. Newman supposed that it was a diameter. “What is the shortest bisecting arc of a square?” Newman answered that it was an altitude through the center. Finally Klamkin asked, “And what is the shortest bisecting arc of an equilateral triangle?”
“By this time, Newman had suspected that I was setting him up (and I was) and almost was going to say the angle bisector,” Klamkin writes. “But he hesitated and said let me consider a chord parallel to the base and since this turns out to be shorter than an angle bisector, he gave this as his answer.”
Was he right?
Proizvolov’s Identity
List the first 2N positive integers (here let N = 4):
1, 2, 3, 4, 5, 6, 7, 8
Divide them arbitrarily into two groups of N numbers:
1, 4, 6, 7
2, 3, 5, 8
Arrange one group in ascending order, the other in descending order:
1, 4, 6, 7
8, 5, 3, 2
Now the sum of the absolute differences of these pairs will always equal N2:
| 1 – 8 | + | 4 – 5 | + | 6 – 3 | + | 7 – 2 | = 16 = N2
(Presented by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Student Olympiads.)
Self-Replicating Resistors
From Lee Sallows:
In an electrical network, if resistors x and y are placed in series their total resistance is x + y; if they’re placed in parallel it’s 1/(1/x + 1/y).
This offers an intriguing opportunity for self-reference. Each of the networks above contains four resistors with values 1, 2, 3, and 4, and the total resistances of the networks themselves are 1, 2, 3, and 4. So any one of the numbered resistors in these networks can be replaced by one of the networks themselves.
The challenge was posed by Sallows and Stan Wagon as a Macalester College “problem of the week”; these examples were discovered by Brian Trial, an automotive electronics engineer from Ferndale, Mich. Sallows points out that any such solution has a dual that results from changing series connections to parallel, and vice versa, and then replacing all resistors values by their reciprocals.
This leads to a further idea: The two sets of resistors below are “co-replicating” — the four networks on the left can be used to replace the four resistors in any of the networks on the right, and vice versa.
(Thanks, Lee.)
The Simson Line
Draw any triangle ABC and pick any point P on its circumcircle.
The closest points to P on lines AB, BC, and AC will be collinear.
Stage Business
The goal of the Shakespeare programming language is to create code that reads like a Shakespearean play: Variables are “characters” that interact through dialogue, constants are represented by nouns and adjectives, and if/then statements are phrased as questions. (Insulting Macbeth assigns him a negative value.) Act and scene numbers serve as GOTO labels, and characters can tell one another to “remember” or “recall” values. The phrases “Open your heart” and “Speak your mind” output a variable’s numerical value and the corresponding ASCII character, respectively.
This program prints the phrase HELLO WORLD:
Romeo, a young man with a remarkable patience.
Juliet, a likewise young woman of remarkable grace.
Ophelia, a remarkable woman much in dispute with Hamlet.
Hamlet, the flatterer of Andersen Insulting A/S.
Act I: Hamlet's insults and flattery.
Scene I: The insulting of Romeo.
[Enter Hamlet and Romeo]
Hamlet:
You lying stupid fatherless big smelly half-witted coward! You are as
stupid as the difference between a handsome rich brave hero and thyself!
Speak your mind!
You are as brave as the sum of your fat little stuffed misused dusty
old rotten codpiece and a beautiful fair warm peaceful sunny summer's
day. You are as healthy as the difference between the sum of the
sweetest reddest rose and my father and yourself! Speak your mind!
You are as cowardly as the sum of yourself and the difference
between a big mighty proud kingdom and a horse. Speak your mind.
Speak your mind!
[Exit Romeo]
Scene II: The praising of Juliet.
[Enter Juliet]
Hamlet:
Thou art as sweet as the sum of the sum of Romeo and his horse and his
black cat! Speak thy mind!
[Exit Juliet]
Scene III: The praising of Ophelia.
[Enter Ophelia]
Hamlet:
Thou art as lovely as the product of a large rural town and my amazing
bottomless embroidered purse. Speak thy mind!
Thou art as loving as the product of the bluest clearest sweetest sky
and the sum of a squirrel and a white horse. Thou art as beautiful as
the difference between Juliet and thyself. Speak thy mind!
[Exeunt Ophelia and Hamlet]
Act II: Behind Hamlet's back.
Scene I: Romeo and Juliet's conversation.
[Enter Romeo and Juliet]
Romeo:
Speak your mind. You are as worried as the sum of yourself and the
difference between my small smooth hamster and my nose. Speak your
mind!
Juliet:
Speak YOUR mind! You are as bad as Hamlet! You are as small as the
difference between the square of the difference between my little pony
and your big hairy hound and the cube of your sorry little
codpiece. Speak your mind!
[Exit Romeo]
Scene II: Juliet and Ophelia's conversation.
[Enter Ophelia]
Juliet:
Thou art as good as the quotient between Romeo and the sum of a small
furry animal and a leech. Speak your mind!
Ophelia:
Thou art as disgusting as the quotient between Romeo and twice the
difference between a mistletoe and an oozing infected blister! Speak
your mind!
[Exeunt]
Because it’s written as a play, a program can be performed by human actors, but the drama lacks a certain narrative drive:
See Output.
Quarters
From Lancelot Hogben’s Mathematics in the Making, an appealingly memorizable table of basic trigonometric values:
See Alison’s Triangle.
(Thanks, Tom.)
Sociable Numbers
The sum of the proper divisors of 14316 is 19116.
The sum of the proper divisors of 19116 is 31704.
The sum of the proper divisors of 31704 is 47616.
The sum of the proper divisors of 47616 is 83328.
The sum of the proper divisors of 83328 is 177792.
The sum of the proper divisors of 177792 is 295488.
The sum of the proper divisors of 295488 is 629072.
The sum of the proper divisors of 629072 is 589786.
The sum of the proper divisors of 589786 is 294896.
The sum of the proper divisors of 294896 is 358336.
The sum of the proper divisors of 358336 is 418904.
The sum of the proper divisors of 418904 is 366556.
The sum of the proper divisors of 366556 is 274924.
The sum of the proper divisors of 274924 is 275444.
The sum of the proper divisors of 275444 is 243760.
The sum of the proper divisors of 243760 is 376736.
The sum of the proper divisors of 376736 is 381028.
The sum of the proper divisors of 381028 is 285778.
The sum of the proper divisors of 285778 is 152990.
The sum of the proper divisors of 152990 is 122410.
The sum of the proper divisors of 122410 is 97946.
The sum of the proper divisors of 97946 is 48976.
The sum of the proper divisors of 48976 is 45946.
The sum of the proper divisors of 45946 is 22976.
The sum of the proper divisors of 22976 is 22744.
The sum of the proper divisors of 22744 is 19916.
The sum of the proper divisors of 19916 is 17716.
The sum of the proper divisors of 17716 is 14316 again.
Rich Earth
Image: Wikimedia Commons
Why do the elements ytterbium, yttrium, terbium, and erbium have similar names?
Because all four of them were first discovered in ore from the same mine near the Swedish village of Ytterby.
Holmium, thulium, and gadolinium were discovered at the same source — leading some to call Ytterby the Galápagos of the periodic table.
Flexagons
Create a strip of 19 triangles like the one above (printable version here) and fold the left portion back successively at each of the northeast-pointing lines to produce a spiral:
Image: Wikimedia Commons
Fold this spiral backward along line ab:
Image: Wikimedia Commons
Then fold the resulting figure backward at cd. You should be left with one blank triangular tab that can be folded backward and pasted to another blank panel on the opposite side. The resulting hexagon should have six 1s on one side and six 2s on the other.
With some adroit pinching this hexagon produces some marvelous effects. Fold down two adjacent triangles so that they meet, and then press in the opposite corner to join them. Now the top of the figure can be prised open and folded down to produce a new hexagon — this one with 1s on one face and a surprising blank on the second. What has become of the 2s?
Exploring the properties of this “hexahexaflexagon” offers an intuitive lesson in geometric group theory:
When Martin Gardner wrote about these bemusing creatures in his first column for Scientific American in 1956, he received two letters. The first was from Neil Uptegrove of Allen B. Du Mont Laboratories in Clifton, N.J.:
Sirs:
I was quite taken with the article entitled ‘Flexagons’ in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder.
But we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in one of the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.
We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of the hexahexaflexagon.
Here is our question: Does his widow draw workmen’s compensation for the duration of his absence, or can we have him declared legally dead immediately? We await your advice.
The second was from Robert M. Hill of The Royal College of Science and Technology in Glasgow, Scotland:
Sirs:
The letter in the March issue of your magazine complaining of the disappearance of a fellow from the Allen B. Du Mont Laboratories ‘down’ a hexahexaflexagon, has solved a mystery for us.
One day, while idly flexing our latest hexahexaflexagon, we were confounded to find that it was producing a strip of multicolored material. Further flexing of the hexahexaflexagon finally disgorged a gum-chewing stranger.
Unfortunately he was in a weak state and, owing to an apparent loss of memory, unable to give any account of how he came to be with us. His health has now been restored on our national diet of porridge, haggis and whisky, and he has become quite a pet around the department, answering to the name of Eccles.
Our problem is, should we now return him and, if so, by what method? Unfortunately Eccles now cringes at the very sight of a hexahexaflexagon and absolutely refuses to ‘flex.’
Saving Face
Image: Wikimedia Commons
“The Joker,” a picture-preserving geomagic square by Lee Sallows. The 16 pieces can be assembled in varying groups of 4 to produce the same picture in 16 different ways, without rotation or reflection.
The outline need not be a joker — it can take almost any shape.
Referee
After seeing a manuscript of On the Origin of Species in April 1859, the Rev. Whitwell Elwin suggested that Darwin write about pigeons instead.
“This appears to me an admirable suggestion,” he wrote. “Everybody is interested in pigeons.”
