Author Archives: Krilling for Company

The Wyrm Ferns

by in Aesthetics, Dragon Curves, Fractals, Geometry, Mathematics and tagged , , , , , , | Leave a comment

A fern is a fractal, a shape that contains copies of itself at smaller and smaller scales. That is, part of a fern looks like the fern as a whole:

Fern as fractal (source)

Millions of years after Mother Nature, man got in on the fract, as it were:

The Sierpiński triangle, a 2d fractal

The Sierpiński triangle is a fractal created in two dimensions by a point jumping halfway towards one or another of the three vertices of a triangle. And here is a fractal created in one dimension by a point jumping halfway towards one or another of the two ends of a line:

A 1d fractal

In one dimension, the fractality of the fractal isn’t obvious. But you can try draggin’ out (or dragon out) the fractality of the fractal by ferning the wyrm, as it were. Suppose that after the point jumps halfway towards one or another of the two points, it’s rotated by some angle around the midpoint of the two original points. When you do that, the fractal becomes more and more obvious. In fact, it becomes what’s called a dragon curve (in Old English, “dragon” was wyrm or worm):

Fractal with angle = 5°

Fractal 10°

Fractal 15°

Fractal 20°

Fractal 25°

Fractal 30°

Fractal 35°

Fractal 40°

Fractal 45°

Fractal 50°

Fractal 55°

Fractal 60°

Fractal 0° to 60° (animated at ezGif)

But as the angle gets bigger, an interesting aesthetic question arises. When is the ferned wyrm, the dragon curve, at its most attractive? I’d say it’s when angle ≈ 55°:

Fractal 50°

Fractal 51°

Fractal 52°

Fractal 53°

Fractal 54°

Fractal 55°

Fractal 56°

Fractal 57°

Fractal 58°

Fractal 59°

Fractal 60°

Fractal 50° to 60° (animated)

At angle >= 57°, I think the dragon curve starts to look like some species of bristleworm, which are interesting but unattractive marine worms:

A bristleworm, Nereis virens (see polychaete at Wikipedia)

Finally, here’s what the ferned wyrm looks like in black-and-white and when it’s rotating:

Fractal 0° to 60° (b&w, animated)

Fractal 56° (rotating)

Fractal 56° (b&w, rotating)

Double fractal 56° (b&w, rotating)

Previously Pre-Posted (Please Peruse)…

• Curvous Energy — a first look at dragon curves
• Back to Drac’ — another look at dragon curves

Toxic Turntable #30

by in Currently Listening..., Music and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Currently listening…

• Vrocsec, Rosa sub Luna (1981)
• Usward Quenched, Trust the Dust (2006)
• Under the Willows, Of Mouse and Man (1999)
• Doom Quota, A Gloomier Land Never Was (2009)
• Jay Victor Caldwell, Symphony in V Minor (1926)
• Elementic TS, Eight’s Too Late (2022)
• Gauntlet Fox, Evensongs of Eleven Counties (1971)
• Les Xenonymphes, Acétone (1995)
• Gnosthrill, God Gnose (1998)
• Malodious, Τῶν Βδελυγμάτων τῆς Γῆς (2007)
• Yickthraite, Om Gom Nom (1997)
• Koukog, Gluehouse (1997)
• Harold Meistmeyer, Best Of (1986)
• Uzuzuzu, We Want Wonders (1992)
• Kotzu, Zone of Clones (1985)
• Liam Tolloway, Ragtime Rex (1913)
• Ptosis, 1991 (1992)
• Nsurosus, Eight Cold Moons (1975)
• Rita Haunts Rita, Ghost to Ghost (2017)

Previously pre-posted:

Toxic Turntable #1 • #2 • #3 • #4 • #5 • #6 • #7 • #8 • #9 • #10 • #11 • #12 • #13 • #14 • #15 • #16 • #17 • #18 • #19 • #20 • #21 • #22 • #23 • #24 • #25 • #26 • #27 • #28 • #29 •

Graph durch Euler

by in Arithmetic, Mathematics, Prime numbers and tagged , , , , , , , , , , , , , | Leave a comment

This is the famous Ulam spiral, in which prime numbers are represented on filled squares on a square spiral:

The Ulam spiral

I like the way the spiral sits between chaos and calm. It’s not wholly random and it’s not wholly regular — it’s betwixt and between. You get a similar chaos-and-calm vibe from a graph for a function called Euler phi. And primes are at work there too. Here’s the graph from Wikipedia:

Graph of eulerphi(n) = φ(n) (see Euler’s totient function)

But what is the Euler phi function? For any integer n, eulerphi(n) gives you the count of numbers < n that are relatively prime to n. That is, the count of numbers < n that have no common factors with n other than one. You can see how eulerphi(n) works by considering whether you can simplify the fraction a/b, where a = 1..n-1 and b = n:

φ(6) = 2
1/6 (1)
2/6 → 1/3
3/6 → 1/2
4/6 → 2/3
5/6, ∴ φ(6) = 2

φ(7) = 6
1/7 (1)
2/7 (2)
3/7 (3)
4/7 (4)
5/7 (5)
6/7, ∴ φ(7) = 6

φ(12) = 4
1/12 (1)
2/12 → 1/6
3/12 → 1/4
4/12 → 1/3
5/12 (2)
6/12 → 1/2
7/12 (3)
8/12 → 2/3
9/12 → 3/4
10/12 → 5/6
11/12, ∴ φ(12) = 4

φ(13) = 12
1/13 (1)
2/13 (2)
3/13 (3)
4/13 (4)
5/13 (5)
6/13 (6)
7/13 (7)
8/13 (8)
9/13 (9)
10/13 (10)
11/13 (11)
12/13, ∴ φ(13) = 12

As you can see, eulerphi(n) = n-1 for primes. Now you know what the top line of the Eulerphi graph is. It’s the primes. Here’s a bigger version of the graph:

Graph of eulerphi(n) = φ(n)

Unlike the Ulam spiral, however, the Eulerphi graph is cramped. But it’s easy to stretch it. You can represent φ(n) as a fraction between 0 and 1 like this: phifrac(n) = φ(n) / (n-1). Using phifrac(n), you can create Eulerphi bands, like this:

Eulerphi band, n <= 1781

Eulerphi band, n <= 3561

Eulerphi band, n <= 7121

Eulerphi band, n <= 14241

Or you can create Eulerphi discs, like this:

Eulerphi disc, n <= 1601

Eulerphi disc, n <= 3201

Eulerphi disc, n <= 6401

Eulerphi disc, n <= 12802

Eulerphi disc, n <= 25602

But what is the bottom line of the Eulerphi bands and inner ring of the Eulerphi discs, where φ(n) is smallest relative to n? Well, the top line or outer ring is the primes and the bottom line or inner ring is the primorials (and their multiples). The function primorial(n) is the multiple of the first n primes:

primorial(1) = 2
primorial(2) = 2*3 = 6
primorial(3) = 2*3*5 = 30
primorial(4) = 2*3*5*7 = 210
primorial(5) = 2*3*5*7*11 = 2310
primorial(6) = 2*3*5*7*11*13 = 30030
primorial(7) = 2*3*5*7*11*13*17 = 510510
primorial(8) = 2*3*5*7*11*13*17*19 = 9699690
primorial(9) = 2*3*5*7*11*13*17*19*23 = 223092870
primorial(10) = 2*3*5*7*11*13*17*19*23*29 = 6469693230

Here are the numbers returning record lows for φfrac(n) = φ(n) / (n-1):

φ(4) = 2 (2/3 = 0.666…)
4 = 2^2
φ(6) = 2 (2/5 = 0.4)
6 = 2.3
φ(12) = 4 (4/11 = 0.363636…)
12 = 2^2.3
[…]
φ(30) = 8 (8/29 = 0.275862…)
30 = 2.3.5
φ(60) = 16 (16/59 = 0.27118…)
60 = 2^2.3.5
[…]
φ(210) = 48 (48/209 = 0.229665…)
210 = 2.3.5.7
φ(420) = 96 (96/419 = 0.2291169…)
420 = 2^2.3.5.7
φ(630) = 144 (144/629 = 0.228934…)
630 = 2.3^2.5.7
[…]
φ(2310) = 480 (480/2309 = 0.2078822…)
2310 = 2.3.5.7.11
φ(4620) = 960 (960/4619 = 0.20783719…)
4620 = 2^2.3.5.7.11
[…]
30030 = 2.3.5.7.11.13
φ(60060) = 11520 (11520/60059 = 0.191811385…)
60060 = 2^2.3.5.7.11.13
φ(90090) = 17280 (17280/90089 = 0.1918103209…)
90090 = 2.3^2.5.7.11.13
[…]
φ(510510) = 92160 (92160/510509 = 0.18052571061…)
510510 = 2.3.5.7.11.13.17
φ(1021020) = 184320 (184320/1021019 = 0.18052553…)
1021020 = 2^2.3.5.7.11.13.17
φ(1531530) = 276480 (276480/1531529 = 0.180525474868579…)
1531530 = 2.3^2.5.7.11.13.17
φ(2042040) = 368640 (368640/2042039 = 0.18052544540040616…)
2042040 = 2^3.3.5.7.11.13.17

Maven of Mixcegenation

by in English Usage, Guardianistas, Mixed Metaphors, Politics and tagged , , , , | Leave a comment

The obfuscating and intentional doublespeak swirling around the emotive cauldron ingredients of “immigration”, “illegal immigration” and “small boats” has been intentionally leveraged into mainstream political and media jargon by Reform UK, big tech algorithms, and thence into the baying mob. […] We are daily enriched by, and should feel deeply indebted to, the many people of colour in this and other sectors of our society. — “This capitulation to racist rhetoric will not end well for Labour or Britain”, letter by Quentin Cowen of Laxfield, Suffolk in The Guardian, 18xi25

Post-Performative Post-Scriptum

“The obfuscating and intentional doublespeak swirling around the emotive cauldron of…” woulda bin even betterer. If the ingredients aren’t bubbling away in the emotive cauldron, why would doublespeak bother to swirl around them? It certainly wouldn’t swirl around them as much, one would’ve thought. And does “emotive cauldron ingredients” mean “emotive-cauldron ingredients” or “emotive cauldron-ingredients”? Maybe it’s both. I’m also struck by the implications of “intentionally leveraged”. Is it possible to “unintentionally leverage” something? Not in this context, one would have thought. And if doublespeak is swirling, that is, if it’s fluid, it’s hard to see how one could exert leverage on it.

Etc, etc. Like all the best Guardianese, this passage is passionately pregnant with interrogation-inducing imagery in a way that is very difficult to achieve by conscious effort. Perhaps Quentin has been smoking some wacky baccy or other psychoactive stimulant supplied by one of the many Persons of Colour enriching his life and fighting da power in da extensive hoodz of Laxfield, Suffolk.

Talking Stalking…

by in Über-Transgression..., Cats and tagged , , , , , , , , , , | Leave a comment

“Most of the trouble in the world has been caused by ten to twenty percent of folks who can’t mind their own business, because they have no business of their own to mind, any more than a smallpox virus.” — Bill Burroughs

“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes like stars start from their spheres,
Thy knotted and combined locks to part,
And each particular hair to stand on end
Like quills upon the fretful porpentine.
But this eternal blazon must not be
To ears of flesh and blood.
List, list, O list!” — Bill Shakespeare

• Aldapuerta’s Acute Angst… A Toxic True Tale of Traumatic Teratotropism…

Hue Views

by in Aesthetics, Art, Color, Quotations and tagged , , , , , , | Leave a comment

The fact is, we none of us enough appreciate the nobleness and sacredness of color. Nothing is more common than to hear it spoken of as a subordinate beauty, — nay, even as the mere source of a sensual pleasure; and we might almost believe that we were daily among men who

“Could strip, for aught the prospect yields
To them, their verdure from the fields;
And take the radiance from the clouds
With which the sun his setting shrouds.”

But it is not so. Such expressions are used for the most part in thoughtlessness; and if the speakers would only take the pains to imagine what the world and their own existence would become, if the blue were taken from the sky, and the gold from the sunshine, and the verdure from the leaves, and the crimson from the blood which is the life of man, the flush from the cheek, the darkness from the eye, the radiance from the hair, — if they could but see for an instant, white human creatures living in a white world, — they would soon feel what they owe to color. The fact is, that, of all God’s gifts to the sight of man, color is the holiest, the most divine, the most solemn. We speak rashly of gay color, and sad color, for color cannot at once be good and gay. All good color is in some degree pensive, the loveliest is melancholy, and the purest and most thoughtful minds are those which love color the most.

• John Ruskin, The Stones of Venice, Vol II, Chapter 5, xxx

Primal Polynomial

by in Arithmetic, Mathematics, Prime numbers and tagged , , , | Leave a comment

n² + n + 17 is one of the best-known polynomial formulas for primes. Its values for n = 0 to 15 are all prime, starting with 17 and ending with 257. — David Wells in The Penguin Dictionary of Curious and Interesting Numbers (1986), entry for “17”

• 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257