Nine Lives

https://commons.wikimedia.org/wiki/File:Arnold%27s_cat_map.png
Image: Wikimedia Commons
https://commons.wikimedia.org/wiki/File:Arnold_cat.png
Image: Wikimedia Commons

In the 1960s, Soviet mathematician Vladimir Arnold mapped the square image of a cat to a torus, “stretched” (sheared) it as shown on that surface, then sliced the resulting image into pieces and recomposed them into a square.

As the process is repeated, any two points in the image quickly become separated, but, surprisingly, after sufficient repetitions the original image reappears.

A discrete analogue is at right. As the transformation is repeated, the image appears increasingly random or disordered, but the underlying cat can be glimpsed making occasional appearances, sometimes as a ghostly suggestion, sometimes in multiple smaller images, and occasionally (yowling, one imagines) even upside down.

It reappears again, unhurt, at the 300th iteration.

It’s called Arnold’s cat map. You can try it yourself here.

| Science & Math

False Fronts

Numbers 23 and 24 Leinster Gardens, London, are not the terraced houses they appear to be. Their façades match their neighbors’, with columned porches and windows and balustraded balconies, but the doors have no knobs or letter boxes.

In fact the whole expanse is false, only a façade 5 feet thick. Behind it is a section of uncovered railway track. When a tube line was built through the neighborhood in 1863, the steam engines that hauled the trains needed a section of uncovered track to let off smoke and steam. To preserve appearances for the surrounding residents, the railway company built these frontages, and they remain to this day.

Writes Stuart Barton in Monumental Follies, “It is unfortunate that more sham facades like this are not built to conceal some of the eye-sores that scar our cities today.”

| Oddities

The Liberal Paradox

If we’re committed to individual rights, then we want a society in which freedoms are not arbitrarily restricted; you should get to exercise your preferences so long as this doesn’t harm someone else. But consider a tiny society of only two people, Lewd and Prude, and a copy of Lady Chatterley’s Lover. Lewd would prefer to read the book rather than have it disposed of unread, but even more than that he’d love to see Prude forced to read it:

Prude reads > Lewd reads > no one reads

Prude would like to see the book disposed of but, short of this, would prefer reading it himself to seeing Lewd enjoy it:

no one reads > Prude reads > Lewd reads

How should a social planner rank these three outcomes? If we’re determined to respect individual rights, then Lewd and Prude should each get to decide whether to read the book; Lewd shouldn’t be prevented from reading it, but Prude shouldn’t be forced to. So we’ll let Lewd’s preference decide between the outcomes “Lewd reads” and “no one reads,” and we’ll let Prude’s decide between “Prude reads” and “no one reads.” But that gives

Lewd reads > no one reads

and

no one reads > Prude reads,

so, for consistency, the social planner arrives at the ranking

Lewd reads > no one reads > Prude reads

and gives the book to Lewd to read.

But both Lewd and Prude would have preferred that Prude be given the book! Harvard economist Amartya Sen argues that this challenges the notion that markets can allocate resources efficiently while respecting individual freedoms.

(Amartya Sen, “The Impossibility of a Paretian Liberal,” Journal of Political Economy 78:1 [1970], 152-157.)

| Society

Snug

https://commons.wikimedia.org/wiki/File:La_maison_du_Gouffre_Plougrescant.jpg
Image: Wikimedia Commons

There’s not much to say about this, but it’s wonderfully picturesque: Standing near the village of Plougrescant on the coast of Brittany is Castel Meur, “the house between the rocks.”

Constructed in 1861, it’s situated between granite rocks to protect it from storms.

01/28/2020 UPDATES:

https://commons.wikimedia.org/wiki/File:1568_-_240411_-_Parte_posterior_de_la_Casa_do_Penedo.jpg
Image: Wikimedia Commons

Casa do Penedo, in northern Portugal, has boulders for its foundation, walls, and ceiling. (Thanks, Rui.)

https://commons.wikimedia.org/wiki/File:Wickhams_Department_Store_(6784618351).jpg
Image: Wikimedia Commons

Wickhams department store planned an imposing new edifice in London’s East End in 1923, but the Spiegelhalter family refused to sell its jewelry shop, so the Wickhams facade had to be built “around” it. The “Spiegelhalter gap” was never closed — the department store closed in the 1960s, while the jewelers’ held out until 1982. (Thanks, Nick.)

st. govan's chapel

St. Govan’s Chapel in Pembrokeshire, Wales, is built into the side of a limestone cliff — visitors must climb down a flight of 52 stairs to reach it. (Thanks, Chris.)

| Oddities

Huaso’s Jump

In February 1949, Chilean army captain Alberto Larraguibel set out to attempt a world-record high jump riding Huaso, a 16-year-old stallion whom he’d trained assiduously in show jumping after undistinguished careers in racing and dressage:

On the first try, I miscalculated the distance and allowed the horse to refuse. If I had then applied the whip, the horse would have become nervous, because an animal understands when it’s being asked to perform above his capabilities. In the second jump, I must have made a mistake of a centimeter or so, because Huaso passed the hands but touched with the belly and the hinds, and knocked down the obstacle … there was only the third and last attempt left. I recalculated again, and in the precise moment we flew… The most difficult moment was the apex of the jump. My eyes were about 4 meters above the ground and I had the sensation of falling head first. My slightest tremor would have been felt by Huaso; who then would have left his hinds behind and we would have crashed together, but we went over. The moment seemed to last forever. I didn’t hear a single shout and thought that something had gone wrong, but I couldn’t hear the obstacles falling either …

Huaso had cleared 2.47 meters (8 feet 1 inch), setting one of the longest-running unbroken sport records in history.

“As for me,” Larraguibel said, “it was like sending my heart flying over the other side of the jump and then going to rescue it.”

| History

Advance Notice

In his 1966 book New Mathematical Diversions From Scientific American, Martin Gardner predicted that the millionth digit of π would be 5. (At the time the value was known only to about 10,000 decimal places.) He was reprinting a column on π from 1960 and included this in the addendum:

It will probably not be long until pi is known to a million decimals. In anticipation of this, Dr. Matrix, the famous numerologist, has sent me a letter asking that I put on record his prediction that the millionth digit of pi will be found to be 5. His calculation is based on the third book of the King James Bible, chapter 14, verse 16 (it mentions the number 7, and the seventh word has five letters), combined with some obscure calculation involving Euler’s constant and the transcendental number e.

He’d intended this as a hoax, but eight years later the computers discovered he was right.

| Science & Math

Sphericons

Fit two identical 90-degrees cones base to base, slice the resulting shape in half vertically, and give one of the halves a quarter turn. The result is a sphericon, a solid that rolls with a bemusing meander: Where the original double cone rolls only in circles, the sphericon puts first one conical sector and then the other in contact with a flat surface beneath it, giving it a smooth but undulating trajectory sustained by a fixed center of mass.

And that’s just the start. “Two sphericons placed next to each other can roll on each other’s surfaces,” writes David Darling in The Universal Book of Mathematics. “Four sphericons arranged in a square block can all roll around one another simultaneously. And eight sphericons can fit on the surface of one sphericon so that any one of the outer solids can roll on the surface of the central one.” See the video for more.

(Thanks, Matthias.)

| Science & Math

The Smithy Code

In deciding a plagiarism case against author Dan Brown in 2006, British justice Peter Smith handed down a peculiar judgment: Certain letters in the text had been italicized with no explanation. Apparently inspired by Brown’s book The Da Vinci Code, Smith had hidden a message in the text.

The judgment included the sentence “The key to solving the conundrum posed by this judgment is in reading HBHG and DVC.” In context, those abbreviations refer to The Holy Blood and the Holy Grail, the book that Brown had been accused of plagiarizing, and The Da Vinci Code.

“I can’t discuss the judgement, but I don’t see why a judgement should not be a matter of fun,” Smith had said in handing down the opinion, which found Brown not guilty. He promised to confirm any correct solution.

He offered enough hints to reporters that Guardian media journalist Daniel Tench eventually solved it: It was a polyalphabetic cipher using a keyword based on the Fibonacci sequence, yielding the plaintext “Jackie Fisher who are you? Dreadnought.” Jackie Fisher was a British admiral whom Smith admired. (The code is described here; Tench describes the solving here.)

The Court of Appeal later said that Smith “was prompted by the extensive use in [The Da Vinci Code] of codes, and no doubt by his own interest in such things, to incorporate a coded message in his judgment, on which nothing turns. The judgment is not easy to read or to understand. It might have been preferable for him to have allowed himself more time for the preparation, checking and revision of the judgment.”

| Puzzles

Podcast Episode 279: The Champawat Tiger

https://vi.m.wikipedia.org/wiki/T%E1%BA%ADp_tin:Maneater_victim.png

At the turn of the 20th century, a rogue tiger terrorized the villages of Nepal and northern India. By the time British hunter Jim Corbett was called in, it had killed 434 people. In this week’s episode of the Futility Closet podcast we’ll describe Corbett’s pursuit of the elusive cat, and his enlightened efforts to address the source of the problem.

We’ll also revisit a Confederate spy and puzzle over a bloody ship.

See full show notes …

| History · Podcast