If three circles “kiss,” like the black circles above, then a fourth circle can be drawn that’s tangent to all three. In 1643 René Descartes showed that if the curvature or “bend” of a circle is defined as k = 1/r, then the radius of the fourth circle can be found by
The ± sign reflects the fact that two solutions are generally possible — the plus sign corresponds to the smaller red circle, the minus sign to the larger (circumscribing) one.
Frederick Soddy summed this up in a poem in Nature (June 20, 1936):
The Kiss Precise
For pairs of lips to kiss maybe
Involves no trigonometry.
‘Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.
In January 1797 London tea broker James Tilly Matthews was committed to the Bethlem psychiatric hospital after increasingly erratic outbursts in which he claimed he was being persecuted by political enemies. In 1809 Matthews’ friends petitioned for his release, arguing that he was no longer insane, and Bethlem apothecary John Haslam published a book-length study showing how bad his case had become.
Matthews believed that a gang of spies were occupying a Roman wall near the asylum and torturing him with a device called an air loom. The loom was operated by “the Middle Man,” while “Sir Archy” and the “Glove Woman” focused its rays on Matthews and “Jack the Schoolmaster” recorded their effects. Similar gangs, Matthews said, were operating looms all over London to influence the thinking of the nation’s leaders. The tortures included “fluid locking,” “cutting soul from sense,” “stone making,” “thigh talking,” and “lobster-cracking,” which Matthews also described as “sudden death-squeezing”:
In short, I do not know any better way for a person to comprehend the general nature of such lobster-cracking operation, than by supposing himself in a sufficiently large pair of nut-crackers or lobster-crackers, with teeth which should pierce as well as press him through every particle within and without; he experiencing the whole stress, torture, driving, oppressing, and crush all together.
Matthews remained an asylum inpatient until his death in 1815. His is now considered to be the first documented case of paranoid schizophrenia.
The programming language Chef, devised by David Morgan-Mar, is designed to make programs look like cooking recipes. Variables are represented by “ingredients,” input comes from the “refrigerator,” output is sent to “baking dishes,” and so on. The language’s design principles state that “program recipes should not only generate valid output, but be easy to prepare and delicious,” but many of them fall short of that goal — one program for soufflé correctly prints the words “Hello world!”, but the recipe requires 32 zucchinis, 101 eggs, and 111 cups of oil to be combined in a bowl and served to a single person. Mike Worth set out to write a working program that could also be read as an actual recipe. Here’s what he came up with:
Hello World Cake with Chocolate sauce. This prints hello world, while being tastier than Hello World Souffle. The main chef makes a " world!" cake, which he puts in the baking dish. When he gets the sous chef to make the "Hello" chocolate sauce, it gets put into the baking dish and then the whole thing is printed when he refrigerates the sauce. When actually cooking, I'm interpreting the chocolate sauce baking dish to be separate from the cake one and Liquify to mean either melt or blend depending on context. Ingredients. 33 g chocolate chips 100 g butter 54 ml double cream 2 pinches baking powder 114 g sugar 111 ml beaten eggs 119 g flour 32 g cocoa powder 0 g cake mixture Cooking time: 25 minutes. Pre-heat oven to 180 degrees Celsius. Method. Put chocolate chips into the mixing bowl. Put butter into the mixing bowl. Put sugar into the mixing bowl. Put beaten eggs into the mixing bowl. Put flour into the mixing bowl. Put baking powder into the mixing bowl. Put cocoa powder into the mixing bowl. Stir the mixing bowl for 1 minute. Combine double cream into the mixing bowl. Stir the mixing bowl for 4 minutes. Liquify the contents of the mixing bowl. Pour contents of the mixing bowl into the baking dish. bake the cake mixture. Wait until baked. Serve with chocolate sauce. chocolate sauce. Ingredients. 111 g sugar 108 ml hot water 108 ml heated double cream 101 g dark chocolate 72 g milk chocolate Method. Clean the mixing bowl. Put sugar into the mixing bowl. Put hot water into the mixing bowl. Put heated double cream into the mixing bowl. dissolve the sugar. agitate the sugar until dissolved. Liquify the dark chocolate. Put dark chocolate into the mixing bowl. Liquify the milk chocolate. Put milk chocolate into the mixing bowl. Liquify contents of the mixing bowl. Pour contents of the mixing bowl into the baking dish. Refrigerate for 1 hour.
Worth confirmed that this correctly prints the words “Hello world!”, and then he used the same instructions to bake a real cake. “It was surprisingly well received,” he writes. “The cake was slightly dry (although nowhere near as dry as cheap supermarket cakes), but this was complimented well by the sauce. My brother even asked me for the recipe!”
While we’re at it: Fibonacci Numbers With Caramel Sauce.
In 1987, paleontologist Tom Rich was leading a dig at Dinosaur Cove southwest of Melbourne when student Helen Wilson asked him what reward she’d get if she found a dinosaur jaw. He said he’d give her a kilo (2.2 pounds) of chocolate. She did, and he did.
Encouraged, the students asked Rich what they’d get if they found a mammal bone. These are fairly rare among dinosaur fossils in Australia, so Rich rashly promised a cubic meter of chocolate — 35 cubic feet, or about a ton.
The cove was “dug out” by 1994, and paleontologists shut down the dig. Rich sent a curious unclassified bone, perhaps a turtle humerus, to two colleagues, who recognized it as belonging to an early echidna, or spiny anteater — a mammal.
Rich now owed the students $10,000 worth of chocolate. “It turns out that it is technically impossible to make a cubic meter of chocolate, because the center would never solidify,” he told National Geographic in 2005. So he arranged for a local Cadbury factory to make a cubic meter of cocoa butter, and then turned the students loose in a room full of chocolate bars.
“It was a bit like Willy Wonka,” Wilson said. “There were chocolate bars on the counters, the tables. We carried out boxes and boxes of chocolate.”
Fittingly, the new echidna was named Kryoryctes cadburyi.
In 1962 a Swedish motorist was fined for leaving his car too long in a space with a posted time limit. The motorist objected, saying that he had removed the car in time and then happened to return to the same spot later, resetting the time limit. The policeman defended his charge, saying that he had noted the positions of the valves on two of the tires — the front-wheel valve was in the 1 o’clock position, the rear-wheel valve at 8 o’clock. If the car had been moved, he argued, the valves were unlikely to take the same positions.
The court accepted the motorist’s claim, calculating that the chance that the valves would return to the same positions by chance was 1/12 × 1/12 = 1/144, great enough to establish reasonable doubt. The court added that if all four valves had been found to be in the same position, the lower likelihood (1/12 × 1/12 × 1/12 × 1/12 = 1/20,736, it figured) would have been enough to uphold the fine.
Is this right? In evaluating this reasoning, University of Chicago law professor Hans Zeisel notes that this method is biased in favor of the defendant, since the positions of the valves are not perfectly independent. He later added, “The use of the 1/144 figure for the probability of the constable’s observations on the assumption that the defendant had driven away also can be questioned. Not only may the rotations of the tires on different axles be correlated, but the figure overlooks the observation that the car was in the same parking spot. When a person leaves a parking place, it is far from certain that the spot will be available later and that the person will choose it again. For this reason, it has been said that the probability of a coincidence is even smaller than a probability involving only the valves.”
(Hans Zeisel, “Dr. Spock and the Case of the Vanishing Women Jurors,” University of Chicago Law Review, 37:1 [Autumn 1969], 1-18)
I am watching a double solar eclipse. The heavenly body Far, traveling east, passes before the sun. Beneath it passes the smaller body Near, traveling west. Far and Near appear to be the same size from my vantage point. Which do I see?
Common sense says that I see Near, since it’s closer. But Washington University philosopher Roy Sorensen argues that in fact I see Far. Near’s existence has no effect on the pattern of light that reaches my eyes. It’s not a cause of what I’m seeing; the view would be the same without it. (Imagine, for example, that Far were much larger and Near was lost in its shadow.)
“When objects are back-lit and are seen by virtue of their silhouettes, the principles of occlusion are reversed,” Sorensen concludes. “In back-lit conditions, I can hide a small suitcase by placing a large suitcase behind it.”
See In the Dark.
(Roy Sorensen, “Seeing Intersecting Eclipses,” Journal of Philosophy XCVI, 1 (1999): 25-49.)
In Other Inquisitions, Borges writes of a strange taxonomy in an ancient Chinese encyclopedia:
On those remote pages it is written that animals are divided into (a) those that belong to the Emperor, (b) embalmed ones, (c) those that are trained, (d) suckling pigs, (e) mermaids, (f) fabulous ones, (g), stray dogs, (h) those that are included in this classification, (i) those that tremble as if they were mad, (j) innumerable ones, (k) those drawn with a very fine camel’s hair brush, (l) others, (m) those that have just broken a flower vase, (n) those that resemble flies from a distance.
This is fanciful, but it has the ring of truth — different cultures can classify the world in surprisingly different ways. In traditional Dyirbal, an aboriginal language of Australia, each noun must be preceded by a variant of one of four words that classify all objects in the universe:
- bayi: men, kangaroos, possums, bats, most snakes, most fishes, some birds, most insects, the moon, storms, rainbows, boomerangs, some spears, etc.
- balan: women, bandicoots, dogs, platypus, echidna, some snakes, some fishes, most birds, fireflies, scorpions, crickets, the hairy mary grub, anything connected with water or fire, sun and stars, shields, some spears, some trees, etc.
- balam: all edible fruit and the plants that bear them, tubers, ferns, honey, cigarettes, wine, cake
- bala: parts of the body, meat, bees, wind, yamsticks, some spears, most trees, grass, mud, stones, noises and language, etc.
“The fact is that people around the world categorize things in ways that both boggle the Western mind and stump Western linguists and anthropologists,” writes UC-Berkeley linguist George Lakoff in Women, Fire, and Dangerous Things (1987). “More often than not, the linguist or anthropologist just throws up his hands and resorts to giving a list — a list that one would not be surprised to find in the writings of Borges.”
adj. producing fever
The 1895 meeting of the Association of American Physicians saw a sobering report: Abraham Jacobi presented the case of a young man whose temperature had reached 149 degrees.
Nonsense, objected William Henry Welch. Such an observation was impossible. He recalled a similar report in the Journal of the American Medical Association (March 31, 1891) in which a Dr. Galbraith of Omaha had found a temperature of 171 degrees in a young woman.
“I do not undertake to explain in what way deception was practised, but there is no doubt in my mind that there was deception,” he said. “Such temperatures as those recorded in Dr. Galbraith’s and Dr. Jacobi’s cases are far above the temperature of heat rigor of mammalian muscle, and are destructive of the life of animal cells.”
Jacobi defended himself: Perhaps medicine simply hadn’t developed a theory to account for such things. But another physician told Welch that Galbraith’s case at least had a perfectly satisfactory explanation — another doctor had caught her in “the old-fashioned trick of heating the thermometer by a hot bottle in the bed.”
These tiles have a remarkable property — by working together, the four can impersonate any one of their number (click to enlarge):
The larger versions could then perform the same trick, and so on. Here’s another set:
In this set, each of the six pieces is paved by some four of them:
By the fathomlessly imaginative Lee Sallows. There’s more in his article “More on Self-Tiling Tile Sets” in last month’s issue of Mathematics Magazine.
Here’s a way to visualize multiplication that reduces it to simple counting:
Express the digits in each factor with rows of parallel lines, as shown, and then count the intersections to derive the product. This is more cumbersome than the traditional method, but its visual nature is appealing, and it permits anyone who can count to reach the right answer even if he doesn’t know the multiplication table.
The example above uses small digits, so no “carrying” is required, but the method does accommodate more complex sums — it’s explained well in this video:
See Two by Two.
A card trick by Mark Wilson:
Put your finger on any red card. Move it left or right to the nearest black card. Move it up or down to the nearest red card. Move it diagonally to the nearest black card. Now move it down or to the right to the nearest red card.
You’ll always land on the ace of diamonds.
(From Harold R. Jacobs, Mathematics: A Human Endeavor, 1970.)
A carpenter named Charlie Bratticks,
Who had a taste for mathematics,
One summer Tuesday, just for fun,
Made a wooden cube side minus one.
Though this to you may well seem wrong,
He made it minus one foot long,
Which meant (I hope your brains aren’t frothing)
Its length was one foot less than nothing,
Its width the same (you’re not asleep?)
And likewise minus one foot deep;
Giving, when multiplied (be solemn!),
Minus one cubic foot of volume.
With sweating brow this cube he sawed
Through areas of solid board;
For though each cut had minus length,
Minus times minus sapped his strength.
A second cube he made, but thus:
This time each one-foot length was plus:
Meaning of course that here one put
For volume, plus one cubic foot.
So now he had, just for his sins,
Two cubes as like as deviant twins:
And feeling one should know the worst,
He placed the second in the first.
One plus, one minus — there’s no doubt
The edges simply canceled out;
So did the volume, nothing gained;
Only the surfaces remained.
Well may you open wide your eyes,
For those were now of double size,
On something which, thanks to his skill,
Took up no room and measured nil.
From solid ebony he’d cut
These bulky cubic objects, but
All that remained was now a thin
Black sharply-angled sort of skin
Of twelve square feet — which though not small,
Weighed nothing, filled no space at all.
It stands there yet on Charlie’s floor;
He can’t think what to use it for!
— J.A. Lindon
How’s that for a headline? It ran in the New York Times Sunday magazine on Aug. 27, 1911:
Canals a thousand miles long and twenty miles wide are simply beyond our comprehension. Even though we are aware of the fact that … a rock which here weighs one hundred pounds would there only weigh thirty-eight pounds, engineering operations being in consequence less arduous than here, yet we can scarcely imagine the inhabitants of Mars capable of accomplishing this Herculean task within the short interval of two years.
The Times was relying on Percival Lowell, who was convinced that a dying Martian civilization was struggling to reach the planet’s ice caps. “The whole thing is wonderfully clear-cut,” he’d told the newspaper — but he was already largely ostracized by skeptical colleagues who couldn’t duplicate his findings. The “spokes” he later saw on Venus may have been blood vessels in his own eye.
Whatever his shortcomings, Lowell’s passions led to some significant accomplishments, including Lowell Observatory and the discovery of Pluto 14 years after his death. “Science,” wrote Emerson, “does not know its debt to imagination.”
A self-reproducing sentence by Lee Sallows — “Doing what it tells you to do yields a replica of itself”:
This reminds me of a short short story by Fredric Brown:
Professor Jones had been working on time theory for many years.
“And I have found the key equation,” he told his daughter one day. “Time is a field. This machine I have made can manipulate, even reverse, that field.”
Pushing a button as he spoke, he said, “This should make time run backward run time make should this,” said he, spoke he as button a pushing.
“Field that, reverse even, manipulate can made have I machine this. Field a is time.” Day one daughter his told he, “Equation key the found have I and.”
Years many for theory time on working been had Jones Professor.
In science it often happens that scientists say, ‘You know, that’s a really good argument; my position is mistaken,’ and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn’t happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.
— Carl Sagan, in a 1987 address, quoted in Jon Fripp et al., Speaking of Science, 2000
On Jan. 9, 1793, two astonished farmers in Woodbury, N.J., watched a strange craft descend from the sky into their field. An excited Frenchman greeted them in broken English and gave them swigs of wine from a bottle. Unable to make himself understood, he finally presented a document:
The farmers helped the man fold his craft and load it onto a wagon for the trip back to Philadelphia. Before leaving, the Frenchman asked them to certify the time and place of his arrival. These details were important — he was Jean-Pierre Blanchard, and he had just completed the first balloon flight in North America.
One hundred sixty-nine years later, when John Glenn went into orbit aboard Friendship 7 in 1962, mission planners weren’t certain where he’d come down. The most likely sites were Australia, the Atlantic Ocean, and New Guinea, but it might be 72 hours before he could be picked up.
Glenn worried about spending three days among aborigines who had seen a silver man emerge from “a big parachute with a little capsule on the end,” so he took with him a short speech rendered phonetically in several languages. It read:
“I am a stranger. I come in peace. Take me to your leader, and there will be a massive reward for you in eternity.”
Theorem 1. A crocodile is longer than it is wide.
Proof. A crocodile is long on the top and bottom, but it is green only on the top; therefore a crocodile is longer than it is green. A crocodile is green along both its length and width, but it is wide only along its width; hence a crocodile is greener than it is wide. Therefore a crocodile is longer than it is wide.
Theorem 2. Napoleon was a poor general.
Proof. Most men have an even number of arms. Napoleon was warned that Wellington would meet him at Waterloo. To be forewarned is to be forearmed. But four arms is a very odd number of arms for a man. The only number that is both even and odd is infinity. Therefore, Napoleon had an infinite number of arms in his battle against Wellington. A general who loses a battle despite having an infinite number of arms is very poor general.
Theorem 3. If 1/0 = ∞, then 1/∞ = 0.
Rotate both sides 90° counterclockwise:
Subtract 8 from both sides:
Now reverse the rotation:
Only the brilliantly inventive Lee Sallows would think of this. The figure above combines Penrose triangles with Borromean rings: Each of the triangles is an impossible object, and they’re united in a perplexing way — although the three are linked together, no two are linked.
Dutch author Leo Lionni devoted most of his career to children’s books, but in 1977 he undertook a weird experiment. Parallel Botany is a catalog of made-up plants, whose made-up features are described by made-up botanists and illustrated by Lionni’s pencil drawings. Sigurya barbulata, at left, is distinguished by its crowning “cephalocarpus”; a specimen discovered in a Mexican pyramid was found to have been metallized into an organic mace, but how this had come about is the subject of “furious debates.”
“The difficulties of applying traditional methods of research to the study of parallel botany stem chiefly from the matterlessness of the plants,” Lionni wrote. “Deprived as they are of any real organs or tissues, their character would be completely indefinable if it were not for the fact that parallel botany is nonetheless botany, and as such it reflects, even if somewhat distantly, many of the most evident features of normal plants.”
Why do all this? Lionni closes with a quote by the made-up Swedish philosopher Erud Kronengaard: “There are two kinds of men, those who are capable of wonder and those who are not. I hope to God that it is the first who will forge our destiny.”
I’m not sure who came up with this — this simple diagram reflects all possible true trigonometric identities of the form x ÷ y = z or x × y = z, where x, y, and z are the basic trigonometric functions of the same angle t.
For any three neighboring functions on the perimeter of the star, the product of the ends always equals the middle (e.g., tan t × cos t = sin t) and the middle function divided by one of the end functions is equal to the other end function (e.g., sin t ÷ tan t = cos t and sin t ÷ cos t = tan t). If you memorize the diagram you can reel off a list of 18 simple relations.
I found it in Michael Stueben’s Twenty Years Before the Blackboard, 1998.
The index to the fourth edition of George Thomas’ Calculus and Analytic Geometry contains an entry for “Whales” on page 188. That page contains no reference to whales, but it does include the figure above.
German mathematician Erich Bessel-Hagen was often teased for his protruding ears.
In 1923 his colleague Béla Kerékjártó published a book, Vorlesungen Über Topologie, whose index lists a reference to Bessel-Hagen on page 151.
That page makes no mention of Bessel-Hagen, but it does contain this figure:
Is that libel?
From Martin Gardner, via Michael Stueben: Obtain a slab of gold measuring 10″ x 11″ x 1″. Divide it diagonally and then cut a triangular notch in two corners as shown. Remove these notches as profit, and slide the remaining halves together to produce a new 10″ x 11″ x 1″ slab. The process can be repeated to yield any amount of money you like!
Kepler’s second law holds that a line segment connecting an orbiting planet to its sun sweeps out equal areas in equal periods of time: In the diagram above, if the time intervals t are equal, then so are the areas A.
If gravity were turned off, would this still be true?