• An der Trauerfeier war im Sinn Nietzsches die sonnige Stille dieser Natureinsamkeit; das Licht spielte durch die Pflaumenbäume an die Kirchmauer und bis in die helle Gruft hinein; eine grosse Spinne spann ihre Gewebe über das Grab von Ästchen zu Ästchen in einem Sonnenstrahl. — Harry Graf Kessler

   • What was Nietzschean in the service was the sunny stillness of this natural solitude: the light playing through the plum trees on the church wall and even in the grave; a large spider spinning her web over the grave from branch to branch in a sunbeam. — Nietzsche is Dead

Get Your Prox Off #2

Serendipity is the art of making happy discoveries by accident. I made a mistake writing a program to create fractals and made the happy discovery of an attractive new fractal. And also of a new version of an attractive fractal I had seen before.

As I described in Get Your Prox Off, you can create a fractal by 1) moving a point towards a randomly chosen vertex of a polygon, but 2) forbidding a move towards the nearest vertex or the second-nearest vertex or third-nearest, and so on. If the polygon is a square, the four possible basic fractals look like this (note that the first fractal is also produced by banning a move towards a vertex that was chosen in the previous move):


v = 4, ban = prox(1)
(ban move towards nearest vertex)


v = 4, ban = prox(2)
(ban move towards second-nearest vertex)


v = 4, ban = prox(3)


v = 4, ban = prox(4)

This program has to calculate what might be called the order of proximity: that is, it creates an array of distances to each vertex, then sorts the array by increasing distance. I was using a bubble-sort, but made a mistake so that the program ran through the array only once and didn’t complete the sort. If this happens, the fractals look like this (note that vertex 1 is on the right, with vertex 2, 3 and 4 clockwise from it):

v = 4, ban = prox(1), sweep = 1


v = 4, ban = prox(2), sweep = 1


v = 4, ban = prox(3), sweep = 1


(Animated version of v4, ban(prox(3)), sw=1)


v = 4, ban = prox(4), sweep = 1

Note that in the last case, where ban = prox(4), a bubble-sort needs only one sweep to identify the most distant vertex, so the fractal looks the same as it does with a complete bubble-sort.

These new fractals looked interesting, so I had the idea of adjusting the number of sweeps in the incomplete bubble-sort: one sweep or two or three and so on (with enough sweeps, the bubble-sort becomes complete, but more sweeps are needed to complete a sort as the number of vertices increases). If there are two sweeps, then ban(prox(1)) and ban(prox(2)) look like this:


v = 4, ban = prox(1), sweep = 2


v = 4, ban = prox(2), sweep = 2

But the fractals produced by sweep = 2 for ban(prox(3)) and ban(prox(4)) are identical to the fractals produced by a complete bubble sort. Now, suppose you add a central point to the polygon and treat that as an additional vertex. If the bubble-sort is incomplete, a ban(prox(1)) fractal with a central point looks like this:


v = 4+c, ban = prox(1), sw = 1


v = 4+c, ban = prox(1), sw = 2

When sweep = 3, an attractive new fractal appears:


v = 4+c, ban = prox(1), sw = 3


v = 4+c, ban = prox(1), sw = 3 (animated)

If you ban two vertices, the nearest and second-nearest, i.e. ban(prox(1), prox(2)), a complete bubble-sort produces a familiar fractal:


v = 4+c, ban = prox(1), prox(2)

And here is ban(prox(2), prox(4)), with a complete bubble-sort:


v = 4, ban = prox(2), prox(4)

If the bubble-sort is incomplete, sweep = 1 and sweep = 2 produce these fractals for ban(prox(1), prox(2)):


v = 4, ban = prox(1), prox(2), sw = 1


v = 4, ban = prox(1), prox(2), sw = 2*

*The second of those fractals is identical to v = 4, ban(prox(2), prox(3)) with a complete bubble-sort.

Here is ban(prox(1), prox(5)) with a complete bubble-sort:


v = 4, ban = prox(1), prox(5)

Now try ban(prox(1), prox(5)) with an incomplete bubble-sort:


v = 4, ban = prox(1), prox(5), sw = 1


v = 4, ban = prox(1), prox(5), sw = 2

When sweep = 3, the fractal I had seen before appears:


v = 4, ban = prox(1), prox(5), sw = 3


v = 4, ban = prox(1), prox(5), sw = 3 (animated)

Where had I seen it before? While investigating this rep-tile (a shape that can be tiled with smaller versions of itself):


L-triomino rep-tile


L-triomino rep-tile (animated)

The rep-tile is technically called an L-triomino, because it looks like a capital L and is one of the two distinct shapes you can create by joining three squares at the edges. You can create fractals from an L-triomino by dividing it into four copies, discarding one of the copies, then repeating the divide-and-discard at smaller and smaller scales:


L-triomino fractal stage #1


L-triomino fractal stage #2


L-triomino fractal stage #3


L-triomino fractal stage #4


L-triomino fractal stage #5


L-triomino fractal (animated)


L-triomino fractal (close-up)

And here’s part of the ban(prox(1), prox(5)) fractal for comparison:



So you can get to the same fractal (or versions of it), by two apparently different routes: random movement of a point inside a square or repeatedly dividing-and-discarding the sub-copies of an L-triomino. That’s serendipity!

Previously pre-posted:

Get Your Prox Off

Toxic Turntable #5

Currently listening…

• Westhenge, Treow (2008)
• Unvit, Swarm ov Static (1987)
• Clangor-DBL, Oculus Omnividens (1994)
• Mallambar, Iesovo (1973)
• QVH, 484 (2011)

Previously pre-posted:

Toxic Turntable #1
Toxic Turntable #2
Toxic Turntable #3
Toxic Turntable #4

Performativizing Papyrocentricity #48

Papyrocentric Performativity Presents:

Vois la ReinePhilip’s Moon Observer’s Guide, Peter Grego (Philip’s 2015)

Gods of FireVolcano Discoveries: A Photographic Journey around the World, Tom Pfeiffer and Ingrid Smet (New Holland 2015)

Chemical TalesRocks and Minerals, Ronald Louis Bonewitz (Dorling Kindersley 2012)

Knyghtes of the RoyalmeMalory: Works, ed. Eugène Vinaver (Oxford University Press 1977)

Alfredo to ZinedineFootball’s Great Heroes and Entertainers, Jimmy Greaves with Norman Giller (Hodder & Stoughton 2007)

Or Read a Review at Random: RaRaR

He Say, He Sigh, He Sow #37

• Il sole, con tutti quei pianeti che girano intorno ad esso e da esso dipendono, può ancora maturare un grappolo d’uva come se non vi fosse nient’altro da fare in tutto l’universo. — Galileo Galilei, 1564-1642.

   • “The sun, with all those planets turning around it and dependent on it, can still ripen a bunch of grapes as if it had nothing else in the universe to do.”