Surface Matters

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In the 17th century, Italian mathematician Evangelista Torricelli experimented with a figure known as Gabriel’s Horn. Rotate the function y = 1/x about the x-axis for x ≥ 1. The resulting figure has finite volume but infinite surface area — it’s sometimes said that, while the horn could be filled up with π cubic units of paint, an infinite number of square units of paint would be needed to cover its surface.

English cosmologist John D. Barrow describes an infinite wedding cake in which each tier is a solid cylinder 1 unit high; the bottom tier has radius 1, the second radius 1/2, the third radius 1/3, and so on. Now the total volume of the cake is π3/6, but the area of its surface is infinite. Barrow writes, “Our infinite cake recipe requires a finite volume of cake to make but it can never be iced because it has an infinite surface area!”

Mike Steuben, a correspondent of Martin Gardner, imagined a set of boxes, each with area 1 × 1. If the height of the first box is 1, the second 1/2, the third 1/4, and so on, then the total volume of the group is 2 cubic units, but the length and the total area of the tops are infinite.

(Barrow’s example is from 100 Essential Things You Didn’t Know You Didn’t Know About Math and the Arts, 2014.)

| Science & Math

Logic and Belief

A syllogism is a logical argument in which a conclusion is inferred from a set of premises:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

The conclusion can be valid without actually stating a true fact; to be valid it just needs to follow logically from the premises. Which of these syllogisms are valid?

No cigarettes are inexpensive.
Some addictive things are inexpensive.
Therefore, some addictive things are not cigarettes.

No addictive things are inexpensive.
Some cigarettes are inexpensive.
Therefore, some cigarettes are not addictive.

In fact both of them are valid. But, interestingly, here the first conclusion seems plausible, while the second does not. That shouldn’t matter, but it does: When Plymouth Polytechnic psychologist J. St. B.T. Evans presented a set of these arguments to subjects in 1983, he found a substantial “belief bias” — the subjects tended to judge the believable conclusions to be valid more than the unbelievable ones. If the conclusion was believable, 92% of the subjects accepted it, regardless of its validity. If the conclusion was unbelievable, 46% accepted it if it was valid, 8% if it was invalid.

Evans wrote, “These findings not only provide a challenge for existing models of syllogistic reasoning but also raise broader questions about people’s rational competence to generate and assess logical arguments in real life, whenever they have clear a priori beliefs about the subject under discussion.”

(J. St. B.T. Evans et al., “On the Conflict Between Logic and Belief in Syllogistic Reasoning,” Memory & Cognition, 11(3), 295-306.)

| Science & Math · Society

Tossing and Turning

Butler University mathematician Jerry Farrell has telekinesis. Here’s a demonstration. Toss a coin and enter the result (HEAD or TAIL) as 1 Across in the grid below. Then solve the rest of the puzzle:

farrell puzzle

Across                                   Down

1 Your coin shows a ______               1 Half a laugh
5 Wagner's earth goddess                 2 Station terminus?
6 Word with one or green                 3 Dec follower?
                                         4 Certain male
Click for Answer
| Puzzles

Unquote

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“There are very few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning; and when they cannot it’s a sign our knowledge of them is very small and confus’d; and when a Mathematical Reasoning can be had it’s as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you.” — John Arbuthnot, Of the Laws of Chance, 1692

| Quotations

As You Wish

In 1951 G.V. Carey published a 15-page booklet called “Making an Index,” intended to guide new authors in preparing indexes for their books. When it was published, a friendly reviewer in the Times Literary Supplement suggested jokingly that the booklet might have benefited from an index of its own, in which Carey could have given “a full-dress demonstration of his principles.”

So, charmingly, Carey made one: In the second edition he added a 3-page index to his 15-page book, writing, “The reviewer, though he may have had his tongue in his cheek, has put the author on his mettle and tempted him, at the opportunity afforded by a new impression, to take up the challenge.” Admittedly, this required some stretching, particularly as he wanted to include every letter of the alphabet. Some sample entries:

Anybody, mere page-numbers not of the slightest use to, 7
Bibliographer, seventeenth-century, 3
Cherry, twice bitten, once shy. See Cross-references
Common sense, use your, 9, 15, and pass.
Earl of Beaconsfield, 11
Eye in, getting your, 5
Fiction, non-, 3
Haystack, looking for needle in, 4
Jehu (son of Nimshi), 12-13
John, St, 10
Life of Abraham Lincoln, 6
Lincoln, Abraham, Life of, 6
Omniscient, indexers not always, 4
Perfection, counsel of, 3
Sense, common. See Common sense
Suez Crisis, 14
What not to do. See Anybody, Earl of Beaconsfield, von Kluck, etc., etc.
York, New, missing, 10
Yourself in the users’ place, put, 6-7, 12
Zealand, New, 10

One thicket of cross-references never finds its way back to the text:

Chase, wild goose, See Von Kluck
Goose chase, wild. See Kluck, von
Kluck, von. See Von Kluck
Von Kluck. See Kluck, von
Wild goose chase. See Kluck, von

And evidently he hates the word alphabetisation:

Order, alphabetical. See Horrid word
Horrid word. See Alphabetisation
Alphabetisation, 9-10

But “It remains only to affirm that the author has made a serious attempt to demonstrate, even in this not very serious index, some at least of the principles set forth in the preceding pages.”

| Literature

A New Pangram

British recreational mathematician Lee Sallows has produced many varieties of the self-enumerating pangram — sentences that inventory their own contents:

This pangram contains four As, one B, two Cs, one D, thirty Es, six Fs, five Gs, seven Hs, eleven Is, one J, one K, two Ls, two Ms, eighteen Ns, fifteen Os, two Ps, one Q, five Rs, twenty-seven Ss, eighteen Ts, two Us, seven Vs, eight Ws, two Xs, three Ys, & one Z.

A few years ago he began to wonder whether it’s possible to produce a sentence that reckons its totals as percentages. This is more difficult, because the percentages won’t always work out to be integers. As he worked on the problem he mentioned it to a few others, among them British computer scientist Chris Patuzzo. And a few days ago, Patuzzo sent him this:

This sentence is dedicated to Lee Sallows and to within one decimal place four point five percent of the letters in this sentence are a’s, zero point one percent are b’s, four point three percent are c’s, zero point nine percent are d’s, twenty point one percent are e’s, one point five percent are f’s, zero point four percent are g’s, one point five percent are h’s, six point eight percent are i’s, zero point one percent are j’s, zero point one percent are k’s, one point one percent are l’s, zero point three percent are m’s, twelve point one percent are n’s, eight point one percent are o’s, seven point three percent are p’s, zero point one percent are q’s, nine point nine percent are r’s, five point six percent are s’s, nine point nine percent are t’s, zero point seven percent are u’s, one point four percent are v’s, zero point seven percent are w’s, zero point five percent are x’s, zero point three percent are y’s and one point six percent are z’s.

Details are here. The next challenge is a version where the percentages are accurate to two decimal places — Patuzzo is working on that now.

(Thanks, Lee.)

| Language · Science & Math

Podcast Episode 81: The Typhus Hoax

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In 1939, as Germany was sending the people of Poland to labor and death camps, two doctors found a unique way to save their countrymen — by faking an epidemic. In this week’s episode of the Futility Closet podcast we’ll learn about their clever plan, which ultimately saved 8,000 people.

We’ll also consider four schemes involving tiny plots of land and puzzle over why a library would waive its fees for a lost book.

See full show notes …

| History · Hoaxes · Podcast

Animal Behavior

In 1949 neurophysiologist Grey Walter built two robot “tortoises” to show that complexity could arise out a very simple nervous system. “Elmer” and “Elsie” each had a light sensor, a touch sensor, a propulsion motor, a steering motor, and two electronic valve-based “neurons.” He found that even with this modest equipment they were capable of phototaxis, finding their way to a recharging station when their batteries ran low. In a subsequent experiment he watched as a robot moved in front of a mirror and responded to its own reflection. “It began flickering,” he wrote. “Twittering, and jigging like a clumsy Narcissus.” He argued that if this behavior were observed in an animal it “might be accepted as evidence of some degree of self-awareness.”

He found that other simple robots were capable of Pavlovian conditioning. When a robot had been taught to seek its “food” near a stool in the middle of the floor, Walter took to blowing a police whistle and kicking the robot before it found the target. “After it had been whistled at and kicked about a dozen times, it learned that a whistle meant trouble. We then removed the specific stimulus — the stool. The whistle was blown, and it avoided the place as if there were a stool there.”

He advanced to a two-note whistle: One pitch was sounded before the robot touched an object, to associate it with avoidance. The other was sounded before it found its food, to associate it with appetite. “The effect of giving both notes was almost always disastrous; it went right off into the darkness on the right-hand side of the room and hovered round there for five minutes in a sort of sulk. It became irresponsive to stimulation and ran round in circles.”

“As you would expect, there are only three ways of alleviating this condition. One of them is rest; in this case that was sufficient, it was left alone to play around in the dark until the effect of all the trauma had died down and it found its way home in the end. Another method is shock, to turn the circuits right off and start again with a clean bill. The most satisfactory method for my purpose is surgery, to dissect out the circuit.”

(Philip Husbands, et al., The Mechanical Mind in History, 2008, and J.M. Tanner & B. Inhelder, eds., Discussions on Child Development, 1958.)

| Technology

The Demon-Haunted World

Index entries from The Golden Bough, Sir James Frazer’s history of myth and religion:

Africa, North, charms to render bridegroom impotent in
Africa, South, disposal of cut hair and nails in; magic use of spittle in; story of the external soul in
Anointing stones, in order to avert bullets from absent warriors; in a rain-charm
Apple-tree, barren women roll under, to obtain offspring; straw man placed on oldest; torches thrown at; as life-index of boys
Bag, souls of persons deposited in a
Beating a man’s garments instead of the man; frogs, as a rain-charm
Birds, cause headache through clipped hair; absent warriors called
Charms, to prevent the sun from going down
Chastity observed for sake of absent persons; as a virtue not understood by savages
Clothes, magic sympathy between a person and his
Conception in women caused by trees
Continence, required during search for sacred cactus
Departmental kings of nature
Dogs crowned
East Indies, pregnant women forbdden to tie knots
Fairies, averse to iron
Fish, magical image to procure
Foreskins used in rainmaking
Gout, transferred to trees
Hyaenas, supposed power over men’s shadow
Impregnation of women by the sun
Jar, the evil of a whole year shut up in
Lemon, external souls of ogres in
Magnets thought to keep brothers at unity
Toothache, transferred to enemies
Twins, water poured on graves of
Whale’s ghost, fear of injuring

Augustus De Morgan wrote, “My opinion of mankind is founded upon the mournful fact that, so far as I can see, they find within themselves the means of believing in a thousand times as much as there is to believe in, judging by experience.”

| Literature