Quick Thinking

https://pixabay.com/en/natural-starry-sky-night-view-2065714/

The historian Socrates tells us that the Emperor Tiberius, who was much given to astrology, used to put the masters of that art, whom he thought of consulting, to a severe test. He took them to the top of his house, and if he saw any reason to suspect their skill, threw them down the steep. Thither he took Thrasyllus, and after a long consultation with him, the emperor suddenly asked whether the astrologer had examined his own fate, and what was portended for him in the immediate future. Now the difficulty is this: If Thrasyllus says that nothing important is about to befall him, he will prove his lack of skill and lose his life besides. If, on the other hand, he says that he is soon to die, either the emperor will set him free, in which case the prophecy was false and he ought to have destroyed him; or Tiberius will destroy him, while he ought to have spared him as a true revealer of the future. Of course the solution is easy. Thrasyllus, after some observations and calculations, began to quake and tremble greatly, and said some great calamity seemed to be impending over him, whereupon the emperor embraced him and made him his chief astrologer.

The Ladies’ Repository, July 1873

Goodbye to Romance

https://books.google.com/books?id=qpctAQAAMAAJ&pg=PA119

In his 1916 book The Science of Musical Sounds, Dayton Clarence Miller uses harmonic analysis to convert the line of a woman’s profile (left) into an equation of 18 terms. Then he uses this equation to reproduce the profile synthetically (right). “If mentality, beauty, and other characteristics can be considered as represented in a profile portrait,” he writes, “then it may be said that they are also expressed in the equation of the profile.”

He repeats the synthesized profile to produce a waveform:

https://books.google.com/books?id=qpctAQAAMAAJ&pg=PA120

“In this sense beauty of form may be likened to beauty of tone color, that is, to the beauty of a certain harmonious blending of sounds.”

In Noise, Water Meat: A History of Sound in the Arts, Douglas Kahn writes, “The simple beauty of the female expressed in the line thus becomes also the simple beauty of mathematics, graphic representation, and instrumentation, let alone mediation and reproduction, involved in the production of the equation and profile. Thus, we move beyond Lord Kelvin’s fascination with a beauty of mathematics to a fascination with a mathematics of beauty.”

Bertrand’s Paradox

https://commons.wikimedia.org/wiki/File:Bertrand.jpg

We ask for the probability that a number, integer or fractional, commensurable or incommensurable, randomly chosen between 0 and 100, is greater than 50. The answer seems evident: the number of favourable cases is half the number of possible cases. The probability is 1/2.

Instead of the number, however, we can choose its square. If the number is between 50 and 100, its square will be between 2,500 and 10,000.

The probability that a randomly chosen number between 0 and 10,000 is greater than 2,500 seems evident: the number of favourable cases is three quarters of the number of possible cases. The probability is 3/4.

The two problems are identical. Why are the two answers different?

— Joseph Bertrand, Calcul des probabilités, 1889 (translation by Sorin Bangu)

Through the Looking-Glass

In 2015, to celebrate the 150th anniversary of the publication of Alice’s Adventures in Wonderland, master sculptor Karen Mortillaro created 12 new sculptures, one for each chapter in Lewis Carroll’s masterpiece. Each takes the form of a table topped with an S-cylindrical mirror, with a bronze sculpture on either side. The sculpture that stands before the mirror is anamorphic, so that the curved mirror’s reflection “undistorts” it, giving it meaning:

http://rmm.ludus-opuscula.org/PDF_Files/Mortillaro_AnamorphicSculpture_49_61(4_2015)_low.pdf

“The S-cylindrical mirror is ideal for this project because it allows for the figures on one side of the mirror to be sculpted realistically, while those on the opposite side of the mirror are distorted and unrecognizable,” Mortillaro writes. “The mirror is symbolic of the parallel worlds that Alice might have experienced in her dream state; the world of reality is on one side of the mirror; and the world of illusion is on the mirror’s opposite side.”

Mortillaro’s article appears in the September 2015 issue of Recreational Mathematics Magazine.

Picket Fences

https://commons.wikimedia.org/wiki/File:First_six_triangular_numbers.svg
Image: Wikimedia Commons

A triangular number is one that counts the number of objects in an equilateral triangle, as above:

1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21

Some of these numbers are palindromes, numbers that read the same backward and forward. A few examples are 55, 66, 171, 595, and 666. In 1973, Charles Trigg found that of the triangular numbers less than 151340, 27 are palindromes.

But interestingly, every string of 1s:

1
11
111
1111
11111

… is a palindromic triangular number in base nine. For example:

119 = 9 + 1 = 10
1119 = 92 + 9 + 1 = 91
11119 = 93 + 92 + 9 + 1 = 820
111119 = 94 + 93 + 92 + 9 + 1 = 7381

The pattern continues — all these numbers are triangular.

02/12/2017 UPDATE: Reader Jacob Bandes-Storch sent a visual proof:

“Given a number n in base 9, if we tack a 1 on the right, the resulting number is 9*n + 1. (By shifting over one place to the left, each digit becomes nine times its original value, and then we add 1 in the ones place.) So given a triangular number, there’s probably a way of sticking together 9 copies of it with a single additional unit to form a new triangle. Sure enough:”

bandes-storch proof 1

bandes-storch proof 2

R.I.P. Raymond Smullyan, 1919–2017

Philosopher and logician Raymond Smullyan passed away on Monday. He was 97.

From my notes, here’s a paradox he offered at a Copenhagen self-reference conference in 2002:

Have you heard of the LAA computing company? Do you know what LAA stands for? It stands for ‘lacking an acronym.’

Actually, the above acronym is not paradoxical; it is simply false. I thought of the following variant which is paradoxical — it is the LACA company. Here LACA stands for ‘lacking a correct acronym.’ Assuming that the company has no other acronym, that acronym is easily seen to be true if and only if it is false.

Noted

https://commons.wikimedia.org/wiki/File:Pascaltriangle2.PNG

In Pascal’s triangle, above, the number in each cell is the sum of the two immediately above it.

If you “tilt” the triangle so that each row starts one column to the right of its predecessor, then the column totals produce the Fibonacci sequence:

pascal triangle fibonacci numbers

That’s from Thomas Koshy’s Triangular Arrays With Applications, 2011.

Bonus: Displace the rows still further and they’ll identify prime numbers.

Podcast Episode 140: Ramanujan

https://commons.wikimedia.org/wiki/File:RamanujanCambridge.jpg

In 1913, English mathematician G.H. Hardy received a package from an unknown accounting clerk in India, with nine pages of mathematical results that he found “scarcely possible to believe.” In this week’s episode of the Futility Closet podcast, we’ll follow the unlikely friendship that sprang up between Hardy and Srinivasa Ramanujan, whom Hardy called “the most romantic figure in the recent history of mathematics.”

We’ll also probe Carson McCullers’ heart and puzzle over a well-proportioned amputee.

Intro:

W.H. Hill’s signature was unchanged when inverted.

Room 308 of West Java’s Samudra Beach Hotel is reserved for the Indonesian goddess Nyai Loro Kidul.

Sources for our feature on Srinivasa Ramanujan:

Robert Kanigel, The Man Who Knew Infinity, 1991.

K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, 1998.

S.R. Ranganathan, Ramanujan: The Man and the Mathematician, 1967.

Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and Commentary, 1991.

G.H. Hardy, “The Indian Mathematician Ramanujan,” American Mathematical Monthly 44:3 (March 1937), 137-155.

Gina Kolata, “Remembering a ‘Magical Genius,'” Science 236:4808 (June 19, 1987), 1519-1521.

E.H. Neville, “Srinivasa Ramanujan,” Nature 149:3776 (March 1942), 293.

Bruce C. Berndt, “Srinivasa Ramanujan,” American Scholar 58:2 (Spring 1989), 234-244.

B.M. Srikantia, “Srinivasa Ramanujan,” American Mathematical Monthly 35:5 (May 1928), 241-245.

S.G. Gindikin, “Ramanujan the Phenomenon,” Quantum 8:4 (March/April 1998), 4-9.

“Srinivasa Ramanujan” in Timothy Gowers, June Barrow-Green, and Imre Leader, eds., Princeton Companion to Mathematics, 2010.

“Srinivasa Aiyangar Ramanujan,” MacTutor History of Mathematics (accessed Jan. 22, 2017).

In the photo above, Ramanujan is at center and Hardy is at far right.

Listener mail:

“Myth Debunked: Audrey Hepburn Did Not Work for the Resistance” [in Dutch], Dutch Broadcast Foundation, Nov. 17, 2016.

“Audrey Hepburn’s Son Remembers Her Life,” Larry King Live, CNN, Dec. 24, 2003.

This week’s lateral thinking puzzle was contributed by listener Tyler Rousseau.

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music or via the RSS feed at http://feedpress.me/futilitycloset.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation on the Support Us page of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

AWOL

https://commons.wikimedia.org/wiki/File:Christ_Church,_Oxford_2.JPG

A harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression (so an example is 1/1, 1/2, 1/3, 1/4 …). When tutoring mathematics at Oxford, Charles Dodgson had a favorite example to illustrate this:

According to him, it is (or was) the rule at Christ Church that, if an undergraduate is absent for a night during term-time without leave, he is for the first offence sent down for a term; if he commits the offence a second time, he is sent down for two terms; if a third time, Christ Church knows him no more. This last calamity Dodgson designated as ‘infinite.’ Here, then, the three degrees of punishment may be reckoned as 1, 2, infinity. These three figures represent three terms in an ascending series of Harmonic Progression, being the counterparts of 1, 1/2, 0, which are three terms in a descending Arithmetical Progression.

— Lionel A. Tollemache, “Reminiscences of ‘Lewis Carroll,'” Literature, Feb. 5, 1898