A pleasing little theorem by Dutch mathematician Frans van Schooten:
Inscribe equilateral triangle ABC in a circle. Now, from a point P on that circle, the length of the longest of segments PA, PB, PC equals the sum of the lengths of the other two segments (in this example, the length of segment PA equals the sum of the lengths of PB and PC).
Last words at the guillotine, collected by Daniel Gerould in Guillotine: Its Legend and Lore (1992):
Catching sight of the statue of liberty opposite the scaffold, Madame Roland cried, “Oh, Liberty, what crimes are committed in thy name!”
From Lee Sallows:
A recent contribution to Futility Closet showed an atypical type of 3×3 geometric magic square in which the 4 pieces occupying each of its nine 2×2 subsquares are able to tile the same rectangle. A different square with the same property is seen in the figure here shown, where the nine tiled rectangles appear at right.
As in the earlier example, the square is to be interpreted as if drawn on a torus, the relations among its peripheral cells then being the same as those that result if the square is surrounded with copies of itself, as seen in the following figure showing four such copies, one in each quadrant:
The figure makes it easier to identify the different 2 × 2 subsquares, exactly nine distinct examples of which can be identified. A brief commentary on the square pointed out that the number of ‘magic’ conditions it satisfies is one greater than the eight conditions demanded by a conventional 3 × 3 magic square. Hence the title of the piece, ‘Extra Magic.’
It was while perusing this diagram that an alternative division of the cells into sets of 4 suggested itself. Instead of 2 × 2 subsquares, consider the four cells defined by a cross that can be centered on any chosen cell. The above figure shows a yellow-shaded example, along with a rectangle tiled by its four associated shapes. It is interesting to note that, as before, there are just nine distinct crosses of this kind to be found in a 3 × 3 square. An obvious question thereby prompted was whether or not a new 3 × 3 magic square could be found based upon such crosses rather than 2×2 subsquares? The answer turned out to be yes, but in the process of scrutinizing an initial specimen I noticed that although it embodied nine cross-based sets of 4 rectangle-tiling pieces, as required, it also included a couple of additional rectangle-tiling sets contained within 2 × 2 subsquares. Clearly the maximum number of such surplus sets would be nine, one for each cross, but could a specimen showing nine cross-based and nine subsquare-based rectangle-tiling sets really exist? I lost no time in seeking an answer.
Regrettably, I was unable to find one. However, the figure below shows a close approach to perfection. It is the same 3 × 3 square with which we started, but now shown alongside no less than eight additional rectangles, each of them tiled with a set of 4 pieces belonging to a cross. Note that the missing rectangle is the one belonging to the non-magic central cross, a show of symmetry that seems appropriate.
So whereas a 3 × 3 magic square, numerical or geometric, satisfies at least 8 separate conditions (3 rows + 3 columns + 2 diagonals), the square here shown satisfies no less than eight more.
(Thanks, Lee.)
In 2011, journalist Alex Renton’s 6-year-old daughter Lulu passed him a letter and asked him to see that it reached the addressee:
To God how did you get invented?
From Lulu
He sent the letter to family members, Christian friends, the Scottish Episcopal Church, the Church of Scotland, and the Scottish Catholic Church. None sent a satisfactory reply. Then he sent it to the Anglican Communion and received this response from Rowan Williams, then Archbishop of Canterbury:
Dear Lulu,
Your dad has sent on your letter and asked if I have any answers. It’s a difficult one! But I think God might reply a bit like this –
‘Dear Lulu – Nobody invented me – but lots of people discovered me and were quite surprised. They discovered me when they looked round at the world and thought it was really beautiful or really mysterious and wondered where it came from. They discovered me when they were very very quiet on their own and felt a sort of peace and love they hadn’t expected.
Then they invented ideas about me – some of them sensible and some of them not very sensible. From time to time I sent them some hints – specially in the life of Jesus – to help them get closer to what I’m really like.
But there was nothing and nobody around before me to invent me. Rather like somebody who writes a story in a book, I started making up the story of the world and eventually invented human beings like you who could ask me awkward questions!’
And then he’d send you lots of love and sign off. I know he doesn’t usually write letters, so I have to do the best I can on his behalf. Lots of love from me too.
Archbishop Rowan
Renton read it to Lulu. “It went down well,” he wrote later. “What worked particularly was the idea of ‘God’s story.’
“‘Well?’ I asked when we reached the end. ‘What do you think?’ She thought a little. ‘Well, I have very different ideas. But he has a good one.'”
In his efforts to reform Russian society, Peter the Great once resorted to banning beards. To bring Russian society more in line with Western Europe, in 1698 he began to charge a fee for the privilege of wearing whiskers, ranging from 100 rubles a year for wealthy merchants down to 1 kopek for a peasant entering a city. Police were empowered to shave scofflaws forcibly.
If you paid your tax you were given a “beard token” with a Russian eagle on one side and a beard on the other. One coin bore the legend THE BEARD IS A SUPERFLUOUS BURDEN.
Because Russians generally resented the law, the tokens are quite valuable now. As early as 1845 collector Walter Hawkins wrote, “The national aversion to the origin of this token probably caused their destruction or dispersion, after they had served their purpose for the year, as they are now very rarely to be met with even in Russia.”
In 2003, students from the University of Plymouth placed a computer keyboard in the enclosure of six Celebes crested macaques in the Paignton Zoo in Devon for one month.
They published the result as Notes Towards the Complete Works of Shakespeare.
It’s hard to know what to make of this. “They are very intentional, deliberate, and very dextrous, so they do want to interact with stuff you give them,” offered zoo biologist Vicki Melfi.
But “the monkeys aren’t reducible to a random process,” concluded test designer Geoff Cox. “They get bored and they shit on the keyboard rather than type.”
Charles Ives set himself an impossible problem [in his 114 Songs of 1922]. He wanted to use pitch distance to represent the fact that God is infinitely close to man. But what is an infinitesimally close pitch distance? In the end Ives gave up and left it to the singer to decide. Maybe what Ives wanted was a smallest perceptible pitch difference. There is no standard notation for this.
— Wilfrid Hodges, “The Geometry of Music,” in John Fauvel, ed., Music and Mathematics, 2006
P. Anning noted this curiosity in Scripta Mathematica in 1956 — if the middle digit 1 in both the numerator and denominator of 101010101/110010011 is replaced with any odd number of 1s, then the proportion remains the same. And all of these numbers are palindromes!
Waclaw Sierpinski gives a proof in 250 Problems in Elementary Number Theory (1970).
In 1983, University of British Columbia physicist Lorne Whitehead noted “a simple and dramatic demonstration of exponential growth, as in a nuclear chain reaction.” He determined that one domino can knock down another that’s about half again as large in all dimensions; since the gravitational potential energy of an upright domino is proportional to the fourth power of its size, this means that one tiny domino can set off a graduated chain reaction with impressively thunderous results.
Whitehead’s first domino was less than 10 mm high; he nudged it with a piece of cotton. The resulting chain reaction brought down a 13th domino that was 64 times as tall; an investment of 0.024 microjoules at one end had released 51 joules of energy at the other, an amplification factor of about 2 billion.
Of course, it’s possible to construct impressive chains of graduated dominoes even if they grow less dramatically than this one. Here’s a world record set in the Netherlands in 2009:
(Lorne A. Whitehead, “Domino ‘Chain Reaction,'” American Journal of Physics 51:2 [February 1983], 183.)
To multiply this 58-digit number by 6, just move its last digit to its head.
Surprisingly, no smaller number displays this behavior when multiplied by 6. It’s called a parasitic number.
