Science & Math

New Units

Since Helen’s face launched a thousand ships, Isaac Asimov proposed that one millihelen was the amount of beauty needed to launch a single ship. And one negative helen is the amount of ugliness that will send a thousand ships in the other direction.

When the taciturn Paul Dirac was a fellow at Cambridge, the dons defined the dirac as the smallest measurable amount of conversation — one word per hour.

Robert Millikan was said to be somewhat conceited; a rival suggested that perhaps the kan was a unit of modesty.

And a bruno is 1158 cubic centimeters, the size of the dent in asphalt resulting from the six-story free fall of an upright piano. It’s named after MIT student Charlie Bruno, who proposed the experiment in 1972. The drop has become an MIT tradition; last year students dropped a piano onto another piano:

Lecture Notes

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“While I am describing to you how Nature works, you won’t understand why Nature works that way. But you see, nobody understands that.” — Richard Feynman

“I am no poet, but if you think for yourselves, as I proceed, the facts will form a poem in your minds.” — Michael Faraday

“Now, this case is not very interesting,” said Bell Labs mathematician Peter Winkler during a lecture at Rutgers. “But the reason why it’s not interesting is really interesting, so let me tell you about it.”

Ernest Rutherford addressed the Royal Institution in 1904:

I came into the room, which was half dark, and presently spotted Lord Kelvin in the audience and realised that I was in for trouble at the last part of the speech dealing with the age of the Earth, where my views conflicted with his. To my relief Kelvin fell fast asleep, but as I came to the important point, I saw the old bird sit up, open an eye, and cock a baleful glance at me. Then a sudden inspiration came and I said Lord Kelvin had limited the age of the Earth, provided no new source was discovered. That prophetic utterance referred to what we are now considering tonight, radium! Behold! the old boy beamed upon me.

When Antonie van Leeuwenhoek declined to teach his new methods in microbiology, Leibniz worried that they might be lost. Leeuwenhoek replied, “The professors and students of the University of Leyden were long ago dazzled by my discoveries. They hired three lens grinders to come to teach the students, but what came of it? Nothing, so far as I can judge, for almost all of the courses they teach there are for the purpose of getting money through knowledge or for gaining the respect of the world by showing people how learned you are, and these things have nothing to do with discovering the things that are buried from our eyes.”

Great Minds

boullee newton cenotaph

In 1784, French architect Étienne-Louis Boullée proposed building an enormous cenotaph for Isaac Newton, a cypress-fringed globe 500 feet high. A sarcophagus would rest on a raised catafalque at the bottom of the sphere; by day light would enter through holes pierced in the globe, simulating starlight, and at night a lamp hung in the center would represent the sun.

“I want to situate Newton in the sky,” Boullée wrote. “Sublime mind! Vast and profound genius! Divine being! Newton! Accept the homage of my weak talents. … O Newton! … I conceive the idea of surrounding thee with thy discovery, and thus, somehow, surrounding thee with thyself.”

As far as I can tell, this is unrelated to Thomas Steele’s proposal to enshrine Newton’s house under a stone globe, which came 41 years later. Apparently Newton just inspired globes.

Succinct

In 2011 M.V. Berry et al. published “Can apparent superluminal neutrino speeds be explained as a quantum weak measurement?” in Journal of Physics A: Mathematical and Theoretical.

The abstract read “Probably not.”

In 1978 John C. Doyle published “Guaranteed margins for LQG regulators” in IEEE Transactions on Automatic Control.

The abstract read “There are none.”

(Thanks, Dre.)

The Parity Paradox

Consider a finite list of n statements:

S1: At least one of statements S1-Sn is false.
S2: At least two of statements S1-Sn is false.

Sn-1: At least n-1 of statements S1-Sn is false.
Sn: At least n of statements S1-Sn is false.

Is this a paradox? It depends: The statements form a self-consistent system if n is even, but not if it’s odd.

From Roy T. Cook’s new book Paradoxes — which is dedicated in part to “anyone whom I don’t discuss in this book.”

Misc

  • Alexander Pope was 4 foot 6.
  • SOCIAL INEPTITUDE is an anagram of POTENTIAL SUICIDE.
  • 6! × 7! = 10!
  • Is the correct answer to this question no?
  • “Do something well, and that is quickly enough.” — Baltasar Gracián

The Hypergame Paradox

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Call a game finite if it terminates in finitely many moves. Now consider Hypergame, which has two rules:

  1. The first player names a finite game.
  2. The two players play that game.

Is Hypergame a finite game? It seems so: It consists of a single game-naming move, followed by a subgame with a necessarily finite number of moves. But what if the first player names Hypergame itself as the subgame, and the second player names Hypergame as the sub-sub-game, and so on?

In presenting this question to students and colleagues at Union College, mathematician William Zwicker found that many saw the catch and quickly pointed out that it leads to infinite play, thinking that this settles the matter. But the proof that Hypergame is finite seems sound. “I … have to convince them that mathematicians cannot simply abandon a proof once a counter-example has been found, for if the internal flaw in such a proof cannot be identified then the counterexample threatens the entire edifice of mathematical proof.” What is the answer?

(William S. Zwicker, “Playing Games with Games: The Hypergame Paradox,” American Mathematical Monthly 94:6, 507-514)

Pleasure Gardening

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The 1957 edition of Exotica: Pictorial Cyclopedia of Indoor Plants included a species called Rumandia cocacoliensis of the family Alcoholiaceae.

The description read “Cuba-libre tall, stemless, succulent, with brown-frosty bloom often with lemon flavor; good in summer, keep cool.”

It was indexed without a page number, and disappeared from subsequent editions.

Field Trip

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In the early 1600s, Johannes Kepler wrote a fantasy in which he imagined a journey to the moon:

We congregate in force and seize a man of this sort; all together lifting him from beneath, we carry him aloft. The first getting into motion is very hard on him, for he is twisted and turned just as if, shot from a cannon, we were sailing across mountains and seas. Therefore, he must be put to sleep beforehand, with narcotics and opiates, and he must be arranged, limb by limb, so that the shock will be distributed over the individual members, lest the upper part of his body be carried away from the fundament, or his head be torn from his shoulders. Then comes a new difficulty: terrific cold and difficulty in breathing. The former we counter with our innate power, the latter by means of moistened sponges applied to the nostrils.

Somnium is largely a treatise on lunar astronomy, describing the motions of the planets as observed from the moon. But Kepler also considers the appearance of the moon’s inhabitants, who “wander in hordes over the whole globe in the space of one of their days, some on foot, whereby they far outstrip our camels, some by means of wings, some in boats pursue the fleeing waters, or if a pause of a good many days is necessary, then they creep into caves.” Carl Sagan and Isaac Asimov called it the first work of science fiction.

Report

Full text of “The Unsuccessful Self-Treatment of a Case of ‘Writer’s Block’,” by Dennis Upper, from the Journal of Applied Behavior Analysis, Fall 1974:

“The Skeptic’s Horoscope”

For Capricorn, Aquarius, Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, and Sagittarius:

The coming year is likely to present challenges; these trials are when your true character will show. Trusted friends can provide assistance in particularly pressing situations. Make use of the skills you have to compensate for ones you lack. Your reputation in the future depends on your honesty and integrity this year. Monetary investments will prove risky; inform yourself as much as possible. On the positive side, your chances of winning the lottery have never been greater!

(By Tim Harrod.)

Business as Usual

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On July 1, 1858, the Linnean Society of London heard a joint presentation by Charles Darwin and Alfred Russel Wallace on the theory of evolution by natural selection.

In his annual report the following May, society president Thomas Bell wrote, “The year which has passed has not, indeed, been marked by any of those striking discoveries which at once revolutionize, so to speak, the department of science on which they bear.”

In Brief

In 1962, botanist Reid Moran published a note in the journal Madroño recounting his collection of a bush rue on a mountaintop in Baja California.

The note’s title was “Cneoridium dumosum (Nuttall) Hooker f. Collected March 26, 1960, at an Elevation of About 1450 Meters on Cerro Quemazón, 15 Miles South of Bahía de Los Angeles, Baja California, México, Apparently for a Southeastward Range Extension of Some 140 Miles.”

The text read, “I got it there then.”

This was followed by a 28-line acknowledgment section in which Moran thanked the person who had reviewed the text, his college professors, and the person who had mailed the manuscript.

Misc

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  • SWARTHMORE is an anagram of EARTHWORMS.
  • The sum of the reciprocals of the divisors of any perfect number is 2.
  • We recite at a play and play at a recital.
  • Is sawhorse the past tense of seahorse?
  • “Things ’twas hard to bear ’tis pleasant to recall.” — Seneca

In Book II, Chapter 9, of H.G. Wells’ novel The War of the Worlds, a sentence begins “For a time I stood regarding …” These words contain 3, 1, 4, 1, 5, and 9 letters.

(Thanks, Dheeraj.)

Memorable Experiments

hanging boy experiment

In 1730 Stephen Gray found that an orphan suspended by insulating silk cords could hold an electrostatic charge and attract small objects.

In 1845, C.H.D. Buys Ballot tested the Doppler effect by arranging for an orchestra of trumpeters to play a single sustained note on an open railroad car passing through Utrecht.

In 1746 Jean-Antoine Nollet arranged 200 Carthusian monks in a circle, each linked to his neighbor with an iron wire. Then he connected the circuit to a rudimentary electric battery.

“It is singular,” he noted, “to see the multitude of different gestures, and to hear the instantaneous exclamation of those surprised by the shock.”

Project Management

How can six people be organized into four committees so that each committee has three members, each person belongs to two committees, and no two committees have more than one person in common?

It’s possible to work this out laboriously, but it yields immediately to a geometric insight:

project management diagram

If each line represents a committee and each intersection is a person, then the problem is solved.

Double Dread

Lyssophobia is fear of hydrophobia.

Bon Appétit

  1. It is now true that Clarence will have a cheese omelette for breakfast tomorrow. [Premise]
  2. It is impossible that God should at any time believe what is false, or fail to believe anything that is true. [Premise: divine omniscience]
  3. Therefore, God has always believed that Clarence will have a cheese omelette for breakfast tomorrow. [From 1, 2]
  4. If God has always believed a certain thing, it is not in anyone’s power to bring it about that God has not always believed that thing. [Premise: the unalterability of the past]
  5. Therefore, it is not in Clarence’s power to bring it about that God has not always believed that he would have a cheese omelette for breakfast. [From 3, 4]
  6. It is not possible for it to be true both that God has always believed that Clarence would have a cheese omelette for breakfast, and that he does not in fact have one. [From 2]
  7. Therefore, it is not in Clarence’s power to refrain from having a cheese omelette for breakfast tomorrow. [From 5, 6]

So Clarence’s eating the omelette tomorrow is not an act of free choice.

From William Hasker, God, Time, and Knowledge, quoted in W. Jay Wood, God, 2011.

The Ballot Box Problem

ballot box lattice diagram

In 1878 W. A. Whitworth imagined an election between two candidates. A receives m votes, B receives n votes, and A wins (m>n). If the ballots are cast one at a time, what is the probability that A will lead throughout the voting?

The answer, it turns out, is given by the pleasingly simple formula

ballot box formula

Howard Grossman offered the proof above in 1946. We start at O, where no votes have been cast. Each vote for A moves us one point east and each vote for B moves us one point north until we arrive at E, the final count, (m, n). If A is to lead throughout the contest, then our path must steer consistently east of the diagonal line OD, which represents a tie score. Any path that starts by going north, through (0,1), must cut OD on its way to E.

If any path does touch OD, let it be at C. The group of such paths can be paired off as p and q, reflections of each other in the line OD that meet at C and continue on a common track to E.

This means that the total number of paths that touch OD is twice the number of paths p that start their journey to E by going north. Now, the first segment of any path might be up to m units east or up to n units north, so the proportion of paths that start by going north is n/(m + n), and twice this number is 2n/(m + n). The complementary probability — the probability of a path not touching OD — is (mn)/(m + n).

(It’s interesting to consider what this means. If m = 2n then p = 1/3 — even if A receives twice as many votes as B, it’s still twice as likely that B ties him at some point as that A leads throughout.)

Theory and Practice

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In 1982 Richard Feynman and his friend Tom Van Sant met in Geneva and decided to visit the physics lab at CERN. “There was a giant machine that was going to be rolled into the line of the particle accelerator,” Van Sant remembered later. “The machine was maybe the size of a two-story building, on tracks, with lights and bulbs and dials and scaffolds all around, with men climbing all over it.

“Feynman said, ‘What experiment is this?’

“The director said, ‘Why, this is an experiment to test the charge-change something-or-other under such-and-such circumstances.’ But he stopped suddenly, and he said, ‘I forgot! This is your theory of charge-change, Dr. Feynman! This is an experiment to demonstrate, if we can, your theory of 15 years ago, called so-and-so.’ He was a little embarrassed at having forgotten it.

“Feynman looked at this big machine, and he said, ‘How much does this cost?’ The man said, ‘Thirty-seven million dollars,’ or whatever it was.

“And Feynman said, ‘You don’t trust me?'”

(Quoted in Christopher Sykes, No Ordinary Genius, 1994.)

Pick’s Theorem

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Georg Alexander Pick found a useful way to determine the area of a simple polygon with integer coordinates. If i is the number of lattice points in the interior and b is the number of lattice points on the boundary, then the area is given by

pick's theorem

There are 40 lattice points in the interior of the figure above and 12 on the boundary, so its area is 40 + 12/2 – 1 = 45.

(Thanks, Pål.)

Misc

  • Only humans are allergic to poison ivy.
  • GUNPOWDERY BLACKSMITH uses 20 different letters.
  • New York City has no Wal-Marts.
  • (5/8)2 + 3/8 = (3/8)2 + 5/8
  • “Ignorance of one’s misfortunes is clear gain.” — Euripides

For any four consecutive Fibonacci numbers a, b, c, and d, ad and 2bc form the legs of a Pythagorean triangle and cdab is the hypotenuse.

(Thanks, Katie.)

Some Palindromes

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In the minuet in Haydn’s Symphony No. 47, the orchestra plays the same passage forward, then backward.

When Will Shortz challenged listeners to submit word-level palindromes to National Public Radio’s Weekend Edition Sunday in 1997, Roxanne Abrams offered the poignant Good little student does plan future, but future plan does student little good.

math palindromes

And Connecticut’s Oxoboxo River offers a four-way palindrome — it reads the same forward and backward both on the page and in a mirror placed horizontally above it.

Heart and Soul

pedal triangle theorem

From a point P, drop perpendiculars to the sides of a surrounding triangle. This defines three points; connect those to make a new triangle and drop perpendiculars to its sides. If you continue in this way, the fourth triangle will be similar to the original one.

In 1947, Mary Pedoe memorialized this fact with a poem:

Begin with any point called P
(That all-too-common name for points),
Whence, on three-sided ABC
We drop, to make right-angled joints,
Three several plumb-lines, whence ’tis clear
A new triangle should appear.

A ghostly Phoenix on its nest
Brooding a chick among the ashes,
ABC bears within its breast
A younger ABC (with dashes):
A figure destined, not to burn,
But to be dropped on in its turn.

By going through these motions thrice
We fashion two triangles more,
And call them ABC (dashed twice)
And thrice bedashed, but now we score
A chick indeed! Cry gully, gully!
(One moment! I’ll explain more fully.)

The fourth triangle ABC,
Though decadently small in size,
Presents a form that perfectly
Resembles, e’en to casual eyes
Its first progenitor. They are
In strict proportion similar.

The property generalizes: Not only is the third “pedal triangle” of a triangle similar to the original triangle, but the nth “pedal n-gon” of an n-gon is similar to the original n-gon.