# Performativizing Papyrocentricity #50

Papyrocentric Performativity Presents:

Life LocomotesRestless Creatures: The Story of Life in Ten Movements, Matt Wilkinson (Icon 2016)

Heart of the MotherJourney to the Centre of the Earth: A Scientific Exploration into the Heart of Our Planet, David Whitehouse (Weidenfeld & Nicolson 2015)

LepidopterobibliophiliaBritish Butterflies: A History in Books, David Dunbar (The British Library 2012)

Minimal Manual – Georgisch Wörterbuch, Michael Jelden (Buske 2016)

Or Read a Review at Random: RaRaR

# For Revver and Fevver

This shape reminds me of the feathers on an exotic bird:

(click or open in new window for full size)

(animated version)

The shape is created by reversing the digits of a number, so you could say it involves revvers and fevvers. I discovered it when I was looking at the Halton sequence. It’s a sequence of fractions created according to a simple but interesting rule. The rule works like this: take n in base b, reverse it, and divide reverse(n) by the first power of b that is greater than n.

For example, suppose n = 6 and b = 2. In base 2, 6 = 110 and reverse(110) = 011 = 11 = 3. The first power of 2 that is greater than 6 is 2^3 or 8. Therefore, halton(6) in base 2 equals 3/8. Here is the same procedure applied to n = 1..20:

1: halton(1) = 1/10[2] → 1/2
2: halton(10) = 01/100[2] → 1/4
3: halton(11) = 11/100[2] → 3/4
4: halton(100) = 001/1000[2] → 1/8
5: halton(101) = 101/1000[2] → 5/8
6: halton(110) = 011/1000 → 3/8
7: halton(111) = 111/1000 → 7/8
8: halton(1000) = 0001/10000 → 1/16
9: halton(1001) = 1001/10000 → 9/16
10: halton(1010) = 0101/10000 → 5/16
11: halton(1011) = 1101/10000 → 13/16
12: halton(1100) = 0011/10000 → 3/16
13: halton(1101) = 1011/10000 → 11/16
14: halton(1110) = 0111/10000 → 7/16
15: halton(1111) = 1111/10000 → 15/16
16: halton(10000) = 00001/100000 → 1/32
17: halton(10001) = 10001/100000 → 17/32
18: halton(10010) = 01001/100000 → 9/32
19: halton(10011) = 11001/100000 → 25/32
20: halton(10100) = 00101/100000 → 5/32…

Note that the sequence always produces reduced fractions, i.e. fractions in their lowest possible terms. Once 1/2 has appeared, there is no 2/4, 4/8, 8/16…; once 3/4 has appeared, there is no 6/8, 12/16, 24/32…; and so on. If the fractions are represented as points in the interval [0,1], they look like this:

point = 1/2

point = 1/4

point = 3/4

point = 1/8

point = 5/8

point = 3/8

point = 7/8

(animated line for base = 2, n = 1..63)

It’s apparent that Halton points in base 2 will evenly fill the interval [0,1]. Now compare a Halton sequence in base 3:

1: halton(1) = 1/10[3] → 1/3
2: halton(2) = 2/10[3] → 2/3
3: halton(10) = 01/100[3] → 1/9
4: halton(11) = 11/100[3] → 4/9
5: halton(12) = 21/100[3] → 7/9
6: halton(20) = 02/100 → 2/9
7: halton(21) = 12/100 → 5/9
8: halton(22) = 22/100 → 8/9
9: halton(100) = 001/1000 → 1/27
10: halton(101) = 101/1000 → 10/27
11: halton(102) = 201/1000 → 19/27
12: halton(110) = 011/1000 → 4/27
13: halton(111) = 111/1000 → 13/27
14: halton(112) = 211/1000 → 22/27
15: halton(120) = 021/1000 → 7/27
16: halton(121) = 121/1000 → 16/27
17: halton(122) = 221/1000 → 25/27
18: halton(200) = 002/1000 → 2/27
19: halton(201) = 102/1000 → 11/27
20: halton(202) = 202/1000 → 20/27
21: halton(210) = 012/1000 → 5/27
22: halton(211) = 112/1000 → 14/27
23: halton(212) = 212/1000 → 23/27
24: halton(220) = 022/1000 → 8/27
25: halton(221) = 122/1000 → 17/27
26: halton(222) = 222/1000 → 26/27
27: halton(1000) = 0001/10000 → 1/81
28: halton(1001) = 1001/10000 → 28/81
29: halton(1002) = 2001/10000 → 55/81
30: halton(1010) = 0101/10000 → 10/81

And here is an animated gif representing the Halton sequence in base 3 as points in the interval [0,1]:

Halton points in base 3 also evenly fill the interval [0,1]. What happens if you apply the Halton sequence to a two-dimensional square rather a one-dimensional line? Suppose the bottom left-hand corner of the square has the co-ordinates (0,0) and the top right-hand corner has the co-ordinates (1,1). Find points (x,y) inside the square, with x supplied by the Halton sequence in base 2 and y supplied by the Halton sequence in base 3. The square will gradually fill like this:

x = 1/2, y = 1/3

x = 1/4, y = 2/3

x = 3/4, y = 1/9

x = 1/8, y = 4/9

x = 5/8, y = 7/9

x = 3/8, y = 2/9

x = 7/8, y = 5/9

x = 1/16, y = 8/9

x = 9/16, y = 1/27…

animated square

Read full page: For Revver and Fevver

From Raymond Smullyan’s Logical Labyrinths (2009):

We now visit another knight/knave island on which, like on the ﬁrst one, all knights tell the truth and all knaves lie. But now there is another complication! For some reason, the natives refuse to speak to strangers, but they are willing to answer yes/no questions using a secret sign language that works like this:

Each native carries two cards on his person; one is red and the other is black. One of them means yes and the other means no, but you are not told which color means what. If you ask a yes/no question, the native will ﬂash one of the two cards, but unfortunately, you will not know whether the card means yes or no!

Problem 3.1. Abercrombie, who knew the rules of this island, decided to pay it a visit. He met a native and asked him: “Does a red card signify yes?” The native then showed him a red card.

From this, is it possible to deduce what a red card signiﬁes? Is it possible to deduce whether the native was a knight or a knave?

Problem 3.2. Suppose one wishes to ﬁnd out whether it is a red card or a black card that signiﬁes yes. What simple yes/no question should one ask?

# Terminal Breach

It’s said that, if you hear “in terms of” 23 times in 23 hours on the 23rd of the month, the ghost of William Burroughs will appear and offer you a heroin enema.

I don’t know whether this is true.

Elsewhere other-engageable:

# Rep-tilian Rites

A pentomino is one of the shapes created by laying five squares edge-to-edge. There are twelve of them (not counting reflections) and this is the P-pentomino:

But it’s not just a pentomino, it’s also a rep-tile, or a shape that can divided into smaller copies of itself. There are two ways of doing this (I’ve rotated the pentomino 90° to make the images look better):

Once you’ve divided the shape into four copies, you can divide the copies, then the copies of the copies, and the copies of the copies of the copies, and so on for ever:

And if you’ve got a reptile, you can turn it into a fractal. Simply divide the shape, discard one or more copies, and continue:

Pentomino-based fractal stage 1

Pentomino-based fractal stage 2

Pentomino-based fractal stage 3

Stage 4

Stage 5

Stage 6

Stage 7

Stage 8

Stage 9

Stage 10

Here are some more fractals created using the same divide-and-discard process:

Animated version

Animated version

You can also use variants on a standard rep-tile dissection, like rotating the copies or trying different patterns of dissection at different levels to see what new shapes appear: