“There was a certain edgy excitement to turning on the computer every morning and immediately checking to see what Mark had thrown down in terms of an ideas-gauntlet.” – Simon Reynolds in the foreword to

K-Punk: The Collected and Unpublished Writings of Mark Fisher (2004–2016), edited by Darren Ambrose, Repeater Books 2018.

Elsewhere other-engageable:

• Ex-term-in-ate!

• Don’t Do Dot…

• Prior Analytics

• Spike-U-Like?

draconic, adj. /drəˈkɒnɪk/ pertaining to, or of the nature of, a dragon. [Latin *draco*, *-ōnem*, < Greek δράκων dragon] — *The Oxford English Dictionary*

In Curvous Energy, I looked at the strange, beautiful and complex fractal known as the dragon curve and showed how it can be created from a staid and sedentary square:

A dragon curve

Here are the stages whereby the dragon curve is created from a square. Note how each square at one stage generates a pair of further squares at the next stage:

Dragon curve from squares #1

Dragon curve from squares #2

Dragon curve from squares #3

Dragon curve from squares #4

Dragon curve from squares #5

Dragon curve from squares #6

Dragon curve from squares #7

Dragon curve from squares #8

Dragon curve from squares #9

Dragon curve from squares #10

Dragon curve from squares #11

Dragon curve from squares #12

Dragon curve from squares #13

Dragon curve from squares #14

Dragon curve from squares (animated)

The construction is very easy and there’s no tricky trigonometry, because you can use the vertices and sides of each old square to generate the vertices of the two new squares. But what happens if you use lines rather than squares to generate the dragon curve? You’ll discover that less is more:

Dragon curve from lines #1

Dragon curve from lines #2

Dragon curve from lines #3

Dragon curve from lines #4

Dragon curve from lines #5

Each line at one stage generates a pair of further lines at the next stage, but there’s no simple way to use the original line to generate the new ones. You have to use trigonometry and set the new lines at 45° to the old one. You also have to shrink the new lines by a fixed amount, 1/√2 = 0·70710678118654752… Here are further stages:

Dragon curve from lines #6

Dragon curve from lines #7

Dragon curve from lines #8

Dragon curve from lines #9

Dragon curve from lines #10

Dragon curve from lines #11

Dragon curve from lines #12

Dragon curve from lines #13

Dragon curve from lines #14

Dragon curve from lines (animated)

But once you have a program that can adjust the new lines, you can experiment with new angles. Here’s a dragon curve in which one new line is at an angle of 10°, while the other remains at 45° (after which the full shape is rotated by 180° because it looks better that way):

Dragon curve 10° and 45°

Dragon curve 10° and 45° (animated)

Dragon curve 10° and 45° (coloured)

Here are more examples of dragon curves generated with one line at 45° and the other line at a different angle:

Dragon curve 65°

Dragon curve 65° (anim)

Dragon curve 65° (col)

Dragon curve 80°

Dragon curve 80° (anim)

Dragon curve 80° (col)

Dragon curve 135°

Dragon curve 135° (anim)

Dragon curve 250°

Dragon curve 250° (anim)

Dragon curve 250° (col)

Dragon curve 260°

Dragon curve 260° (anim)

Dragon curve 260° (col)

Dragon curve 340°

Dragon curve 340° (anim)

Dragon curve 340° (col)

Dragon curve 240° and 20°

Dragon curve 240° and 20° (anim)

Dragon curve 240° and 20° (col)

Dragon curve various angles (anim)

Previously pre-posted:

• Curvous Energy — a first look at dragon curves

]]>• Kev Neuys, *Eloquence Foss* (1964)

• Zetic Load, *Loadstar* (1991)

• Vi Jubilatus, *Quinconce* EP (1979)

• Ylikipojga, *Vazhwevac Xviwv* (1991)

• Voïde Thyroïde, *I by the Yellow Door* (1996)

• Weepster, *Glows the Ghost* (1983)

• Nous les Revenants, *L’Iodisme* (1988)

• Jiji è Vgelu, *Live in Rheims* (2016)

• Hex Dwi, *Cats is Simplicity* (1994)

• Quixotic Plovers, *Imagine Us* (2006)

• Rantique + Mizao, *Oklahama + Ecclesia Sanctorum* (split EP) (2008)

• Elizabeth Dobie, *Uppers/Downers* (1981)

• Zed Follows Wye, *Goats in Etruria* (2000)

• Hyssop-773, *Abeoma* (1987)

• Francesco Manfredini, *12 Concerti* (1997)

• Eark, *Xeno-Zoo* (1993)

Previously pre-posted:

• Toxic Turntable #1

• Toxic Turntable #2

• Toxic Turntable #3

• Toxic Turntable #4

• Toxic Turntable #5

• Toxic Turntable #6

• Toxic Turntable #7

• Toxic Turntable #8

• Toxic Turntable #9

• Toxic Turntable #10

• Toxic Turntable #11

• Toxic Turntable #12

• Toxic Turntable #13

• Toxic Turntable #14

• Toxic Turntable #15

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 paranomasia 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).

• « 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

]]>(Click for larger)

]]>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:

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)

]]>

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)]]>

√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.