The Chills, Submarine Bells (1990)
(Source)
Elsewhere Other-Accessible…
• The Chills — official website
Here’s a useless fact that nobody interested in mathematics would ever forget: digsum(fib(2222)) = 2222. That is, if you add the digits of the 2222nd Fibonacci number, you get 2222:
fib(2222) = 104,966,721,620,282,584,734,867,037,988,863,914,269,721,309,244,628,258,918,225,835,217,264,239,539,186,480,867,849,267,122,885,365,019,934,494,625,410,255,045,832,359,715,759,649,385,824,745,506,982,513,773,397,742,803,445,080,995,617,047,976,796,168,678,756,479,470,761,439,513,575,962,955,568,645,505,845,492,393,360,201,582,183,610,207,447,528,637,825,187,188,815,786,270,477,935,419,631,184,553,635,981,047,057,037,341,800,837,414,913,595,584,426,355,208,257,232,868,908,837,817,478,483,039,310,790,967,631,454,123,105,472,742,221,897,397,857,677,674,619,381,961,429,837,434,434,636,098,678,708,225,493,682,469,5612222 = 1 + 0 + 4 + 9 + 6 + 6 + 7 + 2 + 1 + 6 + 2 + 0 + 2 + 8 + 2 + 5 + 8 + 4 + 7 + 3 + 4 + 8 + 6 + 7 + 0 + 3 + 7 + 9 + 8 + 8 + 8 + 6 + 3 + 9 + 1 + 4 + 2 + 6 + 9 + 7 + 2 + 1 + 3 + 0 + 9 + 2 + 4 + 4 + 6 + 2 + 8 + 2 + 5 + 8 + 9 + 1 + 8 + 2 + 2 + 5 + 8 + 3 + 5 + 2 + 1 + 7 + 2 + 6 + 4 + 2 + 3 + 9 + 5 + 3 + 9 + 1 + 8 + 6 + 4 + 8 + 0 + 8 + 6 + 7 + 8 + 4 + 9 + 2 + 6 + 7 + 1 + 2 + 2 + 8 + 8 + 5 + 3 + 6 + 5 + 0 + 1 + 9 + 9 + 3 + 4 + 4 + 9 + 4 + 6 + 2 + 5 + 4 + 1 + 0 + 2 + 5 + 5 + 0 + 4 + 5 + 8 + 3 + 2 + 3 + 5 + 9 + 7 + 1 + 5 + 7 + 5 + 9 + 6 + 4 + 9 + 3 + 8 + 5 + 8 + 2 + 4 + 7 + 4 + 5 + 5 + 0 + 6 + 9 + 8 + 2 + 5 + 1 + 3 + 7 + 7 + 3 + 3 + 9 + 7 + 7 + 4 + 2 + 8 + 0 + 3 + 4 + 4 + 5 + 0 + 8 + 0 + 9 + 9 + 5 + 6 + 1 + 7 + 0 + 4 + 7 + 9 + 7 + 6 + 7 + 9 + 6 + 1 + 6 + 8 + 6 + 7 + 8 + 7 + 5 + 6 + 4 + 7 + 9 + 4 + 7 + 0 + 7 + 6 + 1 + 4 + 3 + 9 + 5 + 1 + 3 + 5 + 7 + 5 + 9 + 6 + 2 + 9 + 5 + 5 + 5 + 6 + 8 + 6 + 4 + 5 + 5 + 0 + 5 + 8 + 4 + 5 + 4 + 9 + 2 + 3 + 9 + 3 + 3 + 6 + 0 + 2 + 0 + 1 + 5 + 8 + 2 + 1 + 8 + 3 + 6 + 1 + 0 + 2 + 0 + 7 + 4 + 4 + 7 + 5 + 2 + 8 + 6 + 3 + 7 + 8 + 2 + 5 + 1 + 8 + 7 + 1 + 8 + 8 + 8 + 1 + 5 + 7 + 8 + 6 + 2 + 7 + 0 + 4 + 7 + 7 + 9 + 3 + 5 + 4 + 1 + 9 + 6 + 3 + 1 + 1 + 8 + 4 + 5 + 5 + 3 + 6 + 3 + 5 + 9 + 8 + 1 + 0 + 4 + 7 + 0 + 5 + 7 + 0 + 3 + 7 + 3 + 4 + 1 + 8 + 0 + 0 + 8 + 3 + 7 + 4 + 1 + 4 + 9 + 1 + 3 + 5 + 9 + 5 + 5 + 8 + 4 + 4 + 2 + 6 + 3 + 5 + 5 + 2 + 0 + 8 + 2 + 5 + 7 + 2 + 3 + 2 + 8 + 6 + 8 + 9 + 0 + 8 + 8 + 3 + 7 + 8 + 1 + 7 + 4 + 7 + 8 + 4 + 8 + 3 + 0 + 3 + 9 + 3 + 1 + 0 + 7 + 9 + 0 + 9 + 6 + 7 + 6 + 3 + 1 + 4 + 5 + 4 + 1 + 2 + 3 + 1 + 0 + 5 + 4 + 7 + 2 + 7 + 4 + 2 + 2 + 2 + 1 + 8 + 9 + 7 + 3 + 9 + 7 + 8 + 5 + 7 + 6 + 7 + 7 + 6 + 7 + 4 + 6 + 1 + 9 + 3 + 8 + 1 + 9 + 6 + 1 + 4 + 2 + 9 + 8 + 3 + 7 + 4 + 3 + 4 + 4 + 3 + 4 + 6 + 3 + 6 + 0 + 9 + 8 + 6 + 7 + 8 + 7 + 0 + 8 + 2 + 2 + 5 + 4 + 9 + 3 + 6 + 8 + 2 + 4 + 6 + 9 + 5 + 6 + 1
Numbers like this, where k = digsum(fib(k)), are rare. And 2222 is almost certainly the last of them. These are the relevant listings at the Online Encyclopedia of Integer Sequences:
0, 1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222 — A020995, Numbers k such that the sum of the digits of Fibonacci(k) is k.0, 1, 5, 55, 1346269, 9227465, 4052739537881, 498454011879264, 1672445759041379840132227567949787325, 18547707689471986212190138521399707760, 619220451666590135228675387863297874269396512... — A067515, Fibonacci numbers with index = digit sum.
At least, they’re rare in base 10. What about other bases? Well, they’re rare in all other bases except one: base 11. When I looked there, I quickly found more than 450 numbers where digsum(fib(k),b=11) = k. So here’s an interesting little problem: Why is base 11 so productive? Or maybe I should say: Φ is base 11 so productive?
• ἐπαινῶ: παντὶ γάρ μοι δοκεῖ δῆλον ὅτι αὕτη γε ἀναγκάζει ψυχὴν εἰς τὸ ἄνω ὁρᾶν καὶ ἀπὸ τῶν ἐνθένδε ἐκεῖσε ἄγει. — Πολιτεία τοῦ Πλᾰ́τωνος
• • For every one, as I think, must see that astronomy compels the soul to look upwards and leads us from this world to another. — Plato’s Republic (c. 375 BC), Book 7, line 529a
How do you get an hourglass from this shape?
Rep-4 L-tromino
In fact, it’s easy. You simply divide the shape into four identical copies of itself, discard one copy, and repeat the process with each of the sub-copies:
Constructing an hourglass (animated)
↓
Hourglass (static)
Here are some more posts about what I call the hourglass fractal:
• The Hourglass Fractal at Overlord of the Über-feral
An exceptionally ingenious anagram by the American mathematician Mike Keith (born 1955):
hydrogen + zirconium + tin + oxygen + rhenium + platinum + tellurium + terbium + nobelium + chromium + iron + cobalt + carbon + aluminum + ruthenium + silicon + ytterbium + hafnium + sodium + selenium + cerium + manganese + osmium + uranium + nickel + praseodymium + erbium + vanadium + thallium + plutonium
↓
iiiiiiiiiiiiiiiiiiiiiiiiiii + uuuuuuuuuuuuuuuuuuuuuuuuuu + mmmmmmmmmmmmmmmmmmmmmmmmmm + nnnnnnnnnnnnnnnnnnnn + eeeeeeeeeeeeeeee + rrrrrrrrrrrrrr + ooooooooooooo + llllllllllll + aaaaaaaaaaaa + tttttttttt + ccccccc + hhhhhh + bbbbbb + ssssss + dddd + yyyy + ggg + ppp + z + f + v + x + k
↓
nitrogen + zinc + rhodium + helium + argon + neptunium + beryllium + bromine + lutetium + boron + calcium + thorium + niobium + lanthanum + mercury + fluorine + bismuth + actinium + silver + cesium + neodymium + magnesium + xenon + samarium + scandium + europium + berkelium + palladium + antimony + thulium
[as chemical names]
1 + 40 + 50 + 8 + 75 + 78 + 52 + 65 + 102 + 24 + 26 + 27 + 6 + 13 + 44 + 14 + 70 + 72 + 11 + 34 + 58 + 25 + 76 + 92 + 28 + 59 + 68 + 23 + 81 + 94
=
1416
=
7 + 30 + 45 + 2 + 18 + 93 + 4 + 35 + 71 + 5 + 20 + 90 + 41 + 57 + 80 + 9 + 83 + 89 + 47 + 55 + 60 + 12 + 54 + 62 + 21 + 63 + 97 + 46 + 51 + 69
[as atomic numbers]
Elsewhere other-accessible…
• Mike Keith — official website (with anagram here)
Here’s a right triangle, where a^2 + b^2 = c^2. But what are the exact values of a, b, and c?
You might be able to guess by eye, but could you prove your guess? Now try the same right triangle tiled with three identical copies of itself:
1-√3-2 triangle as rep3 rep-tile
Now you can prove the exact values of a, b, and c. If the vertical side, a, is 1, then the hypotenuse, c, is 2, because the length that fits once into a fits twice into c. Therefore 2^2 = 1^2 + b^2 → 4 = 1 + b^2 → 4-1 = b^2 → 3 = b^2 → √3 = b. The horizontal side, b, has a length of √3 = 1.73205080757… So the right triangle is 1-√3-2. And if it’s rep3, that is, can be divided into three identical copies of itself, then it’s also rep9, rep27, and so on:
1-√3-2 triangle as rep9 rep-tile
1-√3-2 triangle as rep27 rep-tile
1-√3-2 triangle as rep81 rep-tile
1-√3-2 triangle as rep243 rep-tile
1-√3-2 triangle as rep729 rep-tile
Once you’ve got a rep-tile, you can create fractals. But the 1-√3-2 triangle is cramped. You need more space to work with. And it’s easy to find that space when you realize that a standard equilateral triangle can be divided into six 1-√3-2 triangles:
Equilateral triangle divided into six 1-√3-2 triangles
Equilateral triangle tiled with 1-√3-2 triangles (stage 1)
(please open in new window if image is distorted)
Equilateral triangle tiled with 1-√3-2 triangles (stage 2)
Equilateral triangle tiled with 1-√3-2 triangles (stage 3)
Here are variant colorings of the stage-3 tiled triangle:
But where are the fractals? In one way, you’ve already seen them. But they get more obvious like this:
Fractal stage 1
Fractal #2
Fractal #3
Fractal #4
Fractal #5
Fractal #6
Fractal #7
Fractal #8
Fractal (animated)
Another fractal stage 1
[…]
↓
[…]
Another fractal #8
Another fractal (animated)
And when you have a fractal created using an equilateral triangle, it’s easy to expand the fractal into a circle, like this:
Original fractal
↓
Fractal expanded into circle
Triangular fractal to circular fractal (animated)
ვენერა — რიგით მეორე პლანეტა მზიდან; მისი სიმკვრივე და აგებულება მსგავსია დედამიწისა (Translate.ge)
金星 — 金 jīn, jìn, gold; metals in general; money; 星 xīng, a star, planet; any point of light (MDBG)
Grumpy Dyke’s Phallocrator (2023)
Elsewhere other-accessible…
• Grumpy Dyke at Bandcamp
• R I O A E T L U
→ Herriot a été élu.
• L N N E O P Y I A V Q E I E D C D
→ Hélène est née au pay grec, y a vécu et y est décédée.
• J J A D I D A C K O T L A H E T D B K C D G A L E V D I N C P I E D F I J E C O Q P D B B A J T
→ Gigi a des idées assez cahotées: elle a acheté des bécasses et des geais, a élevé des hyènes, s’est payé des effigies et s’est occupée des bébés agités.
• From John Julius Norwich’s More Christmas Crackers (1990)
Construction of a Sierpiński tetrahedron (from WikiMedia)
Post-Performative Post-Scriptum
The toxic title of this incendiary intervention radically references George Harrison’s album Extra Texture (1975).
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 
