# Monthly Archives: February 2017

# Kea Here

# Performativizing Papyrocentricity #52

Papyrocentric Performativity Presents:

• Reds in the Head – *The War of the Worlds*, H.G. Wells (1898)

• Canine the Barbarian – *The Call of the Wild, White Fang, and Other Stories*, Jack London (Penguin American Library 1981)

• Star-Stuff – *The Universe in 100 Key Discoveries*, Giles Sparrow (Quercus 2012)

• An Island of Her Own – *The Phantom Atlas: The Greatest Myths, Lies and Blunders on Maps*, Edward Brooke-Hitching (Simon & Schuster 2016)

Or Read a Review at Random: RaRaR

# The Swing’s the Thing

Order emerges from chaos with a triangle or pentagon, but not with a square. That is, if you take a triangle or a pentagon, chose a point inside it, then move the point repeatedly halfway towards a vertex chosen at random, a fractal will appear:

Sierpiński triangle from point jumping halfway to randomly chosen vertex

Sierpiński pentagon from point jumping halfway to randomly chosen vertex

But it doesn’t work with a square. Instead, the interior of the square slowly fills with random points:

Square filling with point jumping halfway to randomly chosen vertex

As I showed in Polymorphous Perverticity, you can create fractals from squares and randomly moving points if you ban the point from choosing the same vertex twice in a row, and so on. But there are other ways. You can take the point, move it towards a vertex at random, then swing it around the center of the square through some angle before you mark its position, like this:

Point moves at random, then swings by 90° around center

Point moves at random, then swings by 180° around center

You can also adjust the distance of the point from the center of the square using a formula like

*dist*=

*r**

*rm*–

*dist*, where

*dist*is the distance,

*r*is the radius of the circle in which the circle is drawn, and

*rm*takes values like 0.1, 0.25, 0.5, 0.75 and so on:

Point moves at random, *dist* = *r* * 0.05 – *dist*

Point moves at random, *dist* = *r* * 0.1 – *dist*

Point moves at random, *dist* = *r* * 0.2 – *dist*

But you can swing the point while applying a vertex-ban, like banning the previously chosen vertex, or the vertex 90° or 180° away. In fact, swinging the points converts one kind of vertex ban into the others.

Point moves at random towards vertex not chosen previously

Point moves at random, then swings by 45°

Point moves at random, then swings by 360°

Point moves at random, then swings by 697.5°

Point moves at random, then swings by 720°

Point moves at random, then swings by 652.5°

Animated angle swing

You can also reverse the swing at every second move, swing the point around a vertex instead of the center or around a point on the circle that encloses the square. Here are some of the fractals you get applying these techniques.

Point moves at random, then swings alternately by 45°, -45°

Point moves at random, then swings alternately by 90°, -90°

Point moves at random, then swings alternately by 135°, -135°

Point moves at random, then swings alternately by 180°, -180°

Point moves at random, then swings alternately by 225°, -225°

Point moves at random, then swings alternately by 315°, -315°

Point moves at random, then swings alternately by 360°, -360°

Animated alternate swing

Point moves at random, then swings around point on circle by 45°

Point moves at random, then swings around point on circle by 67.5°

Point moves at random, then swings around point on circle by 90°

Point moves at random, then swings around point on circle by 112.5°

Point moves at random, then swings around point on circle by 135°

Point moves at random, then swings around point on circle by 180°

Animated circle swing

# He Say, He Sigh, He Sow #42

« Il n’y avait plus dans la rue que les boutiquiers et les chats. » — Albert Camus, *L’Étranger* (1942).

“There was no longer anything in the street but shopkeepers and cats.” — Camus, *The Outsider*.

# Tri Again (Again)

I didn’t expect to find the hourglass fractal playing with squares. I even less expected it playing with triangles. Isosceles right triangles, to be precise. Then again, I found it first playing with the L-triomino, which is composed of three squares. And an isosceles triangle is half of a square. So it all fits. This is an isosceles right triangle:

Isosceles right triangle

It’s mirror-symmetrical, so it looks the same in a mirror unless you label one of the acute-angled corners in some way, like this:

Right triangle with labelled corner

Right triangle reflected

Reflection is how you find the hourglass fractal. First, divide a right triangle into four smaller right triangles.

Right triangle rep-tiled

Then discard one of the smaller triangles and repeat. If the acute corners of the smaller triangles have different orientations, one of the permutations creates the hourglass fractal, like this:

Hourglass #1

Hourglass #2

Hourglass #3

Hourglass #4

Hourglass #5

Hourglass #6

Hourglass #7

Hourglass #8

Hourglass #9

Hourglass animated

Another permutation of corners creates what I’ve decided to call the crane fractal, like this:

Crane fractal animated

Crane fractal (static)

The crane fractal is something else that I first found playing with the L-triomino:

Crane fractal from L-triomino

Previously pre-posted: