Rigging in the Trigging

Here’s a simple pattern of three triangles:

Three-Triangle Pattern

Now replace each triangle in the pattern with the same pattern at a smaller scale:

Replacing triangles

If you keep on doing this, you create what I’ll call a trigonal fractal (trigon is Greek for “triangle”):

Trigonal Fractal stage #3 (click for larger)

Trigonal Fractal stage #4

Trigonal Fractal stage #5

Trigonal Fractal #6

Trigonal Fractal #7

Trigonal Fractal #8

Trigonal Fractal (animated) (click for larger)

You can use the same pattern to create different fractals by rotating the replacement patterns in different ways. I call this “rigging the trigging” and here are some of the results:

You can also use a different seed-pattern to create the fractals:

Trigonal fractal (animated)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Trigonal fractal (anim)

Note: The title of this incendiary intervention is of course a paronomasia on the song “Frigging in the Rigging”, also known as “Good Ship Venus” and performed by the Sex Pistols on The Great Rock ’n’ Roll Swindle (1979).

He Say, He Sigh, He Sow #48

• « S’il est un homme tourmenté par la maudite ambition de mettre tout un livre dans une page, toute une page dans une phrase, et tout une phrase dans un mot, c’est moi. » — Joseph Jourbet (1754-1824)

• “If there is a man tormented by the cursed ambition to compress an entire book into a page, an entire page into a phrase, and that phrase into a word, it is I.” — Joseph Jourbet

Curvous Energy

Here is a strange and beautiful fractal known as a dragon curve:

A dragon curve (note: this is a twin-dragon curve or Davis-Knuth dragon)

And here is the shape generally regarded as the dullest and most everyday of all:

A square

But squares are square, so let’s go back to dragon-curves. This particular kind of dragon-curve looks a lot like a Chinese dragon. You can see the same writhing energy and scaliness:

Chinese dragon

Dragon-curve for comparison

Dragon-curves also look like some species of soft coral:

Red soft-coral

In short, dragon-curves are organic and lively, quite unlike the rigid, lifeless solidity of a square. But there’s more to a dragon-curve than immediately meets the eye. Dragon-curves are rep-tiles, that is, you can tile one with smaller copies of itself:

Dragon-curve rep-tiled with two copies of itself

Dragon-curve rep-4

Dragon-curve rep-8

Dragon-curve rep-16

Dragon-curve rep-32

Dragon-curve self-tiling (animated)

From the rep-32 dragon-curve, you can see that a dragon-curve can be surrounded by six copies of itself. Here’s an animation of the process:

Dragon-curve surrounded (anim)

And because dragon-curves are rep-tiles, they will tile the plane:

Dragon-curve tiling #1

Dragon-curve tiling #2

But how do you make these strange and beautiful shapes, with their myriad curves and curlicules, their energy and liveliness? It’s actually very simple. You start with the shape generally regarded as the dullest and most everyday of all:

A square

Then you see how the shape can be replaced by five smaller copies of itself:

Square overlaid by five smaller squares

Square replaced by five smaller squares

Then you set about replacing it with two of those smaller copies:

Replacing squares Stage #0

Replacing squares Stage #1

Then you do it again to each of the copies:

Replacing squares Stage #2

And again:

Replacing squares #3

And again:

Replacing squares #4

And keep on doing it:

Replacing squares #5

Replacing squares #6

Replacing squares #7

Replacing squares #8

Replacing squares #9

Replacing squares #10

Replacing squares #11

Replacing squares #12

Replacing squares #13

Replacing squares #14

Replacing squares #15

And in the end you’ve got a dragon-curve:

Dragon-curve built from squares

Dragon-curve built from squares (animated)

Year and Square

The simplest and in some ways greatest magic square is this:

6 1 8
7 5 3
2 9 4 (Magic total = 15)

All rows and columns sum to 15 and so do both diagonals. Using other sets of numbers, you can create an infinite number of further 3×3 magic squares. Here’s one using only prime numbers and 1:

43 01 67
61 37 13
07 73 31 (Magic=111)

The magic total is 111, which is 3 x 37, just as 15 = 3 x 5. It’s an interesting but untaxing exercise to prove that, for all 3×3 magic squares, the magic total is three times the central number. So you can use only prime numbers in a 3×3 square, but you can’t have a prime number as the magic total (unless you use fractions and so on).

And guess what? 2019 = 3 x 667, the first prime number after 666. So I decided to see if I could find an all-prime magic squares whose magic total was 2019. I found nine of them (and 9 = 3 x 3).

1117 0019 0883
0439 0673 0907
0463 1327 0229 (Magic=2019)

1069 0067 0883
0487 0673 0859
0463 1279 0277 (Magic=2019)

1063 0229 0727
0337 0673 1009
0619 1117 0283 (Magic=2019)

0883 0313 0823
0613 0673 0733
0523 1033 0463 (Magic=2019)

0619 0337 1063
1117 0673 0229
0283 1009 0727 (Magic=2019)

0463 0439 1117
1327 0673 0019
0229 0907 0883 (Magic=2019)

0463 0487 1069
1279 0673 0067
0277 0859 0883 (Magic=2019)

0379 0607 1033
1327 0673 0019
0313 0739 0967 (Magic=2019)

0523 0613 0883
1033 0673 0313
0463 0733 0823 (Magic=2019)

Pi and By

Here’s √2 in base 2:

√2 = 1.01101010000010011110... (base=2)

And in base 3:

√2 = 1.10201122122200121221... (base=3)

And in bases 4, 5, 6, 7, 8, 9 and 10:

√2 = 1.12220021321212133303... (b=4)
√2 = 1.20134202041300003420... (b=5)
√2 = 1.22524531420552332143... (b=6)
√2 = 1.26203454521123261061... (b=7)
√2 = 1.32404746317716746220... (b=8)
√2 = 1.36485805578615303608... (b=9)
√2 = 1.41421356237309504880... (b=10)

And here’s π in the same bases:

π = 11.00100100001111110110... (b=2)
π = 10.01021101222201021100... (b=3)
π = 03.02100333122220202011... (b=4)
π = 03.03232214303343241124... (b=5)
π = 03.05033005141512410523... (b=6)
π = 03.06636514320361341102... (b=7)
π = 03.11037552421026430215... (b=8)
π = 03.12418812407442788645... (b=9)
π = 03.14159265358979323846... (b=10)

Mathematicians know that in all standard bases, the digits of √2 and π go on for ever, without falling into any regular pattern. These numbers aren’t merely irrational but transcedental. But are they also normal? That is, in each base b, do the digits 0 to [b-1] occur with the same frequency 1/b? (In general, a sequence of length l will occur in a normal number with frequency 1/(b^l).) In base 2, are there as many 1s as 0s in the digits of √2 and π? In base 3, are there as many 2s as 1s and 0s? And so on.

It’s a simple question, but so far it’s proved impossible to answer. Another question starts very simple but quickly gets very difficult. Here are the answers so far at the Online Encyclopedia of Integer Sequences (OEIS):

2, 572, 8410815, 59609420837337474 – A049364

The sequence is defined as the “Smallest number that is digitally balanced in all bases 2, 3, … n”. In base 2, the number 2 is 10, which has one 1 and one 0. In bases 2 and 3, 572 = 1000111100 and 210012, respectively. 1000111100 has five 1s and five 0s; 210012 has two 2s, two 1s and two 0s. Here are the numbers of A049364 in the necessary bases:

10 (n=2)
1000111100, 210012 (n=572)
100000000101011010111111, 120211022110200, 200011122333 (n=8410815)
11010011110001100111001111010010010001101011100110000010, 101201112000102222102011202221201100, 3103301213033102101223212002, 1000001111222333324244344 (n=59609420837337474)

But what number, a(6), satisfies the definition for bases 2, 3, 4, 5 and 6? According to the notes at the OEIS, a(6) > 5^434. That means finding a(6) is way beyond the power of present-day computers. But I assume a quantum computer could crack it. And maybe someone will come up with a short-cut or even an algorithm that supplies a(b) for any base b. Either way, I think we’ll get there, π and by.

A Seriously Sizzling Series of Super-Saucy Salvadisms

Some good quotes by Salvador Dalí (1904-89), who will need no introduction to keyly committed core components of the quixotically contrarian community. The Spanish should be reliable, but the English translations may not be (coz i dun em).

• A los seis años quería ser cocinero. A los siete quería ser Napoleón. Mi ambición no ha hecho más que crecer; ahora sólo quiero ser Salvador Dalí y nada más. Por otra parte, esto es muy difícil, ya que, a medida que me acerco a Salvador Dalí, él se aleja de mí.
 — At six years of age I wanted to be a chef. At seven I wanted to be Napoleon. My ambition has only grown since then, but now I only want to be Salvador Dalí and nothing more. Still, it’s very difficult, because the closer I get to Salvador Dalí, the further he gets from me.

• El canibalismo es una de las manifestaciones más evidentes de la ternura.
 — Cannibalism is a sure sign of affection.

• El que quiere interesar a los demás tiene que provocarlos.
 — He who wishes to interest other people needs to provoke them.

• …Es curioso, a mi me interesa mucho mas hablar, o estar en contacto con la gente que piensa lo contrario de lo que yo pienso, que de los que piensan lo mismo que pienso yo.
 — …It’s strange, but I’d much rather talk with or be in touch with people who think the opposite of what I think than with those who think the same as I do.

• Es fácil reconocer si el hombre tiene gusto: la alfombra debe combinar con las cejas.
 — It’s easy to tell if a man has good taste: his carpet should harmonize with his eyebrows.

• De ninguna manera volveré a México. No soporto estar en un país más surrealista que mis pinturas.
 — Under no circumstances will I return to Mexico. I cannot bear to be in a country more surreal than my own paintings.

• Hoy, el gusto por el defecto es tal que sólo parecen geniales las imperfecciones y sobre todo la fealdad. Cuando una Venus se parece a un sapo, los seudoestetas contemporáneos exclaman: ¡Es fuerte, es humano!
 — Today, a taste for the defective is so strong that the only things that seem attractive are imperfections and, above all, ugliness. When a Venus looks like a toad, the pseudo-aesthetes of today shout: “That’s great, that’s human!”

• Los errores tienen casi siempre un carácter sagrado. Nunca intentéis corregirlos. Al contrario: lo que procede es racionalizarlos, compenetrarse con aquellos integralmente. Después, os será posible subliminarlos.
— Mistakes almost always have a sacred character. Never try to correct them. On the contrary, you need to ponder them, to examine them from every angle. Afterwards, you will be able to absorb them.

• La Revolución Rusa es la Revolución Francesa que llega tarde, por culpa del frío.
 — The Russian Revolution is the French Revolution arriving late due to the cold.

• La única diferencia entre un loco y Dalí, es que Dalí no está loco.
 — The only difference between a madman and Dalí is that Dalí is not mad.

• La vida es aspirar, respirar y expirar.
 — Life is aspiring, respiring and expiring.

• Lo importante es que hablen de ti, aunque sea bien.
 — What’s important is that people talk about you, even if they only say good things.

• Lo único de lo que el mundo no se cansará nunca es de la exageración.
 — The only thing the world never tires of is exaggeration.

• ¡No podéis expulsarme porque Yo soy el Surrealismo!
 — You cannot expel me: I am Surrealism! (After being expelled from the surrealist movement in Paris.)

• Picasso es pintor. Yo también. Picasso es español. Yo también. Picasso es comunista. Yo tampoco.
 — Picasso is a painter. So am I. Picasso is a Spaniard. So am I. Picasso is a communist. Nor am I.

• Sin una audiencia, sin la presencia de espectadores, estas joyas no alcanzarían la función para la cual fueron creadas. El espectador, por tanto, es el artista final. Su vista, corazón, mente — con una mayor o menor capacidad para entender la intención del creador — da vida a las joyas.
 — Without an audience, without a circle of spectators, these jewels would never realize the purpose for which they were created. The spectator is therefore the final artist. His eyes, his heart, his mind — whether better or worse equipped to understand the purpose of the creator — give life to the jewels.

• Llamo a mi esposa: Gala, Galuska, Gradiva; Oliva por lo oval de su rostro y el color de su piel; Oliveta, diminutivo de la oliva; y sus delirantes derivados: Oliueta, Oriueta, Buribeta, Buriueteta, Siliueta, Solibubuleta, Oliburibuleta, Ciueta, Liueta. También la llamo Lionette, porque cuando se enfada ruge como el león de la Metro-Goldwyn Mayer.
 — I call my wife Gala, Galuska, Gradiva; Oliva for her oval face and the colour of her skin; Oliveta, diminutive of Oliva; and its delirious derivations: Oliueta, Oriueta, Buribeta, Buriueteta, Siliueta, Solibubuleta, Oliburibuleta, Ciueta, Liueta. I also call her Lionette, because when she’s angry she roars like the MGM lion.

• Sólo hay dos cosas malas que pueden pasarte en la vida, ser Pablo Picasso o no ser Salvador Dalí.
 — There are only two things that can go wrong for you in life: being Pablo Picasso or not being Salvador Dalí.

• Si muero, no moriré del todo.
 — If I die, I will not die completely. (Compare Horace’s Non omnis moriar, I will not wholly die.)

• La inteligencia sin ambición es un pájaro sin alas.
 — Intelligence without ambition is a bird without wings.

• No tengas miedo de la perfección, nunca la alcanzarás.
 — Don’t be afraid of perfection, because you’ll never achieve it.

• Para comprar mis cuadros hay que ser criminalmente rico como los norteamericanos.
 — To buy my paintings you have to be criminally rich like the Americans.

• Hay días en que pienso que voy a morir de una sobredosis de satisfacción.
 — There are days when I think that I will die of an overdose of satisfaction.

• El termómetro del éxito no es más que la envidia de los descontentos.
 — The thermometer of success is nothing more than the envy of the discontent.

• Lo menos que puede pedirse a una escultura es que no se mueva.
 — The least that one can ask of a sculpture is that it stays still.

• Mientras estamos dormidos en este mundo, estamos despiertos en el otro.
 — When we are asleep in this world, we are awake in another.

• Yo no tomo drogas. Yo soy una droga.
 — I do not take drugs. I am a drug.

• Los que no quieren imitar nada, no producen nada.
 — Those who refuse to imitate will never create.

• Las guerras nunca han hecho daño a nadie, excepto a la gente que muere.
 — Wars have never done harm to anyone, except to those who die.

• Gustar el dinero como me gusta, es nada menos que misticismo. El dinero es una gloria.
 — To relish money as I do is nothing short of mysticism. Money is a glory.

• La existencia de la realidad es la cosa más misteriosa, más sublime y más surrealista que se dé.
 — The existence of reality is the most mysterious, most sublime and most surrealist thing of all.

Performativizing Papyrocentricity #66

Papyrocentric Performativity Presents:

Pygmies and Secret PolicemenFootball Against the Enemy, Simon Kuper (1994)

Writhing Along in My AutomobileCrash: The Limits of Car Safety, Nicholas Faith (Boxtree 1998)

A Boy and His BanditBeloved and God: The Story of Hadrian and Antinoüs, Royston Lambert (Weidenfeld & Nicolson 1984)

Or Read a Review at Random: RaRaR