He Say, He Sigh, He Show #33

Apud me omnia fiunt Mathematicè in NaturaRené Descartes (1596-1650)
  • For me, all things in nature occur mathematically. Correspondence with Martin Mersenne (1640).

The Art Grows Onda

Anyone interested in recreational mathematics should seek out three compendiums by Ian Stewart: Professor Stewart’s Cabinet of Mathematical Curiosities (2008), Professor Stewart’s Hoard of Mathematical Treasures (2009) and Professor Stewart’s Casebook of Mathematical Mysteries 2014). They’re full of ideas and puzzles and are excellent introductions to the scope and subtlety of maths. I first came across Alexander’s Horned Sphere in one of them. I also came across this simpler shape that packs infinity into a finite area:

unicorn_triangle

I call it a horned triangle or unicorn triangle and it reminds me of a wave curling over, like Katsushika Hokusai’s The Great Wave off Kanagawa (c. 1830) (“wave” is unda in Latin and onda in Spanish).

The Great Wave off Kanagawa by Katsushika Hokusai (1760–1849)

The Great Wave off Kanagawa by Katsushika Hokusai (1760–1849)

To construct the unicorn triangle, you take an equilateral triangle with sides of length 1 and erect a triangle with sides of length 0.5 on one of its corners. Then on the corresponding corner of the new triangle you erect a triangle with sides of length 0.25. And so on, for ever.

unicorn_multicolor

unicorn_animated

When you double the sides of a polygon, you quadruple the area: a 1×1 square has an area of 1, a 2×2 square has an area of 4. Accordingly, when you halve the sides of a polygon, you quarter the area: a 1×1 square has an area of 1, a 0.5 x 0.5 square has an area of 0.25 or 1/4. So if the original triangle of the unicorn triangle above has an area of 1 rather than sides of 1, the first triangle added has an area of 0.25 = 1/4, the next an area of 0.0625 = 1/16, and so on. The infinite sum is this:

1/4 + 1/16 + 1/256 + 1/1024 + 1/4096 + 1/16384…

Which equals 1/3. This becomes important when you see the use made of the shape in Stewart’s book. The unicorn triangle a rep-tile, or a shape that can be divided into smaller copies of the same shape:

unicorn_reptile_static

unicorn_reptile

An equilateral triangle can be divided into four copies of itself, each 1/4 of the original area. If an equilateral triangle with an area of 4 is divided into three unicorn triangles, each unicorn has an area of 1 + 1/3 and 3 * (1 + 1/3) = 4.

Because it’s a rep-tile, a unicorn triangle is also a fractal, a shape that is self-similar at smaller and smaller scales. When one of the sub-unicorns is dropped, the fractals become more obvious:

unicorn_fractal1


unicorn_fractal2


unicorn_fractal3


Elsewhere other-posted:

Rep-Tiles Revisited

Performativizing Papyrocentricity #45

Papyrocentric Performativity Presents:

Plants on PaperDrawing and Painting Plants, Christina Brodie (A & C Black 2006)

LewminiferousGuide to Garden Wildlife, Richard Lewington (British Wildlife Publishing 2008)

Old GoldPuskás: Madrid, the Magyars and the Amazing Adventures of the World’s Greatest Goalscorer, György Szöllős (Freight Books 2015)

Rosetta RokRok 1984, George Orwell (MUZA SA, Warszawa 2001)


Or Read a Review at Random: RaRaR

Shareway to Seven

An adaptation of an interesting distribution puzzle from Joseph Degrazia’s Math is Fun (1954):

After a successful year of plunder on the high seas, a pirate ship returns to its island base. The pirate chief, who enjoys practical jokes and has a mathematical bent, hands out heavy bags of gold coins to his seven lieutenants. But when the seven lieutenants open the bags, they discover that each of them has received a different number of coins.

They ask the captain why they don’t have equal shares. The pirate chief laughs and tells them to re-distribute the coins according to the following rule: “At each stage, the lieutenant with most coins must give each of his comrades as many coins as that comrade already possesses.”

The lieutenants follow the rule and each one in turn becomes the lieutenant with most coins. When the seventh distribution is over, all seven of them have 128 coins, the coins are fairly distributed, and the rule no longer applies.

The puzzle is this: How did the pirate captain originally allocate the coins to his lieutenants?


If you start at the beginning and work forward, you’ll have to solve a fiendishly complicated set of simultaneous equations. If you start at the end and work backwards, the puzzle will resolve itself almost like magic.

The puzzle is actually about powers of 2, because 128 = 2^7 and when each of six lieutenants receives as many coins as he already has, he doubles his number of coins. Accordingly, before the seventh and final distribution, six of the lieutenants must have had 64 coins and the seventh must have had 128 + 6 * 64 coins = 512 coins.

At the stage before that, five of the lieutenants must have had 32 coins (so that they will have 64 coins after the sixth distribution), one must have had 256 coins (so that he will have 512 coins after the sixth distribution), and one must have had 64 + 5 * 32 + 256 coins = 480 coins. And so on. This is what the solution looks like:

128, 128, 128, 128, 128, 128, 128
64, 64, 64, 64, 64, 64, 512
32, 32, 32, 32, 32, 480, 256
16, 16, 16, 16, 464, 240, 128
8, 8, 8, 456, 232, 120, 64
4, 4, 452, 228, 116, 60, 32
2, 450, 226, 114, 58, 30, 16
449, 225, 113, 57, 29, 15, 8

So the pirate captain must have originally allocated the coins like this: 449, 225, 113, 57, 29, 15, 8.

The puzzle can be adapted to other powers. Suppose the rule runs like this: “At each stage, the lieutenant with most coins must give each of his comrades twice as many coins as that comrade already possesses.” If the pirate captain has six lieutenants, after each distribution each of five will have n + 2n = three times the number of coins that he previously possessed. The six lieutenants each end up with 729 coins = 3^6 coins and the solution looks like this:

729, 729, 729, 729, 729, 729
243, 243, 243, 243, 243, 3159
81, 81, 81, 81, 2997, 1053
27, 27, 27, 2943, 999, 351
9, 9, 2925, 981, 333, 117
3, 2919, 975, 327, 111, 39
2917, 973, 325, 109, 37, 13

For powers of 4, the rule runs like this: “At each stage, the lieutenant with most coins must give each of his comrades three times as many coins as that comrade already possesses.” With five lieutenants, each of them ends up with 1024 coins = 4^5 coins and the solution looks like this:

1024, 1024, 1024, 1024, 1024
256, 256, 256, 256, 4096
64, 64, 64, 3904, 1024
16, 16, 3856, 976, 256
4, 3844, 964, 244, 64
3841, 961, 241, 61, 16

For powers of 5, the rule runs like this: “At each stage, the lieutenant with most coins must give each of his comrades four times as many coins as that comrade already possesses.” With four lieutenants, each of them ends up with 625 coins = 5^4 coins and the solution looks like this:

625, 625, 625, 625
125, 125, 125, 2125
25, 25, 2025, 425
5, 2005, 405, 85
2001, 401, 81, 17

Shick Shtick

Slightly adapted from Joseph Degrazia’s Math is Fun (1954):

Six Writers in a Railway Car

On their way to Chicago for a conference of authors and journalists, six writers meet in a railway club car. Three of them sit on one side facing the other three. Each of the six has his specialty. One writes short stories, one is a historian, another one writes humorous books, still another writes novels, the fifth is a playwright and the last a poet. Their names are Abbott, Blake, Clark, Duggan, Eccles and Farmer.* Each of them has brought one of his books and given it to one of his colleagues, so that each of the six is deep in a book which one of the other five has written.

Abbott reads a collection of short stories. Clark reads the book written by the colleague sitting just opposite him. Blake sits between the author of the short stories and the humorist. The short-story writer sits opposite the historian. Duggan reads a play. Blake is the brother-in-law of the novelist. Eccles sits next to the playwright. Abbott sits in a corner and is not interested in history. Duggan sits opposite the novelist. Eccles reads a humorous book. Farmer never reads poems.

These facts are sufficient to find each of the six authors’ specialties.


*In the original, the surnames were Blank, Bird, Grelly, George, Pinder and Winch.

Performativizing Papyrocentricity #44

Papyrocentric Performativity Presents:

Lesser LettersYou’ve Had Your Time: Being the Second Part of the Confessions of Anthony Burgess, Anthony Burgess (Heinemann 1990)

The Light of DaySJWs Always Lie: Taking Down the Thought Police, Vox Day (Castalia House 2015)

Sextual KeelingSextant: A Voyage Guided by the Stars and the Men Who Mapped the World’s Oceans, David Barrie (William Collins 2014)

Twy Defy the EyeThe World of Visual Illusions: Optical Tricks That Defy Belief!, Gianni A. Sarcone and Marie-Jo Waeber (Arcturus 2012)


Or Read a Review at Random: RaRaR

Don’t Do Dot…

It’s a mistake to think that Guardianese, the optimal dialect of keyly committed core components of the counter-cultural community, mandates optionizing on a permanent basis for the pretentious and polysyllabic. Yes, Guardianistas are addicted to phrases like “in terms of” and “prior to”, but they also like urgently throbbing monosyllables like “key”, “core” and “spike”.

These are unnatural words, taken from headlines, not from normal English. They reveal an important truth: simplicity can be pretentious too. The two aspects of Guardianese come together in phrases like “key indicator” and “core metric”. I would say that “vital sign” and “important statistic” are better and more natural English, but you can’t tell that by counting syllables.

And sometimes Guardianese doesn’t use any syllables at all…  Guardianistas also like the stylistic trick of trailing dots. I find it cheap and irritating, so I’m glad that one of my favourite writers thought the same long ago. In his essay “Stories I Have Tried to Write”, M.R. James (1862-1936) said this:

In parenthesis, many common objects may be made the vehicles of retribution, and where retribution is not called for, of malice. Be careful how you handle the packet you pick up in the carriage-drive, particularly if it contains nail parings and hair. Do not, in any case, bring it into the house. It may not be alone… (Dots are believed by many writers of our day to be a good substitute for effective writing. They are certainly an easy one. Let us have a few more……) (“Stories I Have Tried To Write”, 1929)

In short: Don’t do dot…


Elsewhere other-engageable:

Ex-term-in-ate!
Titus Graun
Reds under the Thread

Mater Mysteriorum

“But just as all of Baltimore pondered the mystery of how a progressive city could produce such a despotic police force, a second mystery had presented itself: If everyone was organized to prevent violence, why did it continue to happen? The cops were back at their posts. The whole city had been politicized. The poorest streets were filled with activist group meetings and sermons. The gangs were professing nonviolence. Still, the murders continued.” — “A Most Violent City”, New York Magazine, viâ Steve Sailer.

Terminal Teraticity…

Americotrop-ism: n. literally, a tendency or habit of moving towards America; hence, a tendency or habit of adopting American culture, vocabulary and usage when one is not oneself American. | -ist, -ic(al) [Americ(a) + -o- + tropism, on the analogy of phototropism, chemotropism, geotropism, etc]

• “The Guardian-reading community displays a marked Americotropism.”