Florence Nightingale created this innovative diagram to track the causes of mortality in the Crimean War. Her description:

The Areas of the blue, red, & black wedges are each measured from the centre as the common vertex.

The blue wedges measured from the centre of the circle represent area for area the deaths from Preventable or Mitigable Zymotic diseases, the red wedges measured from the centre the deaths from wounds, & the black wedges measured from the centre the deaths from all other causes.

The black line across the red triangle in Nov. 1854 marks the boundary of the deaths from all other causes during the month.

In October 1854, & April 1855, the black area coincides with the red, in January & February 1856, the blue coincides with the black.

The entire areas may be compared by following the blue, the red, & the black lines enclosing them.

The diagram on the right corresponds to the first 12 months of the war; the one on the left shows the second 12 months. The difference reflects the dramatic effectiveness of a sanitary commission in reducing disease.

Nightingale found that presenting information graphically made it more accessible to Members of Parliament and civil servants, who might not otherwise read statistical reports. In 1859 she was elected the first female member of the Royal Statistical Society.

In 1982, computer scientists Edward Fredkin and Tommaso Toffoli suggested that it might be possible to construct a computer out of bouncing billiard balls rather than electronic signals. Spherical balls bouncing frictionlessly between buffers and other balls could create circuits that execute logic, at least in principle.

In 2011, Yukio-Pegio Gunji and his colleagues at Kobe University extended this idea in an unexpected direction: They found that “swarms of soldier crabs can implement logical gates when placed in a geometrically constrained environment.” These crabs normally live in lagoons, but at low tide they emerge in swarms that behave in predictable ways. When placed in a corridor and menaced with a shadow representing a crab-eating bird, a swarm will travel forward, and if it encounters another swarm the two will merge and continue in a direction that’s the sum of their respective velocities.

Gunji et al. created a set of corridors that would act as logic gates, first in a simulation and then with groups of 40 real crabs. The OR gate, where two groups of crabs merge, worked well, but the AND gate, which requires the merged swarm to choose one of three paths, was less reliable. Still, the researchers think they can improve this result by making the environment more crab-friendly — which means that someday a working crab-powered computer may be possible.

(Yukio-Pegio Gunji, Yuta Nishiyama, and Andrew Adamatzky, “Robust Soldier Crab Ball Gate,” Complex Systems 20:2 [2011], 93–104.)

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

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:

“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.”

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)

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:”

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.

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:

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

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