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

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For Revver and Fevver

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

feathers

(click or open in new window for full size)


feathers_anim

(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 thann.

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:

line1_1_2

point = 1/2


line2_1_4

point = 1/4


line3_3_4

point = 3/4


line4_1_8

point = 1/8


line5_5_8

point = 5/8


line6_3_8

point = 3/8


line7_7_8

point = 7/8


line_b2_anim

(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]:

line_b3_anim


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:

square1

x = 1/2, y = 1/3


square2

x = 1/4, y = 2/3


square3

x = 3/4, y = 1/9


square4

x = 1/8, y = 4/9


square5

x = 5/8, y = 7/9


square6

x = 3/8, y = 2/9


square7

x = 7/8, y = 5/9


square8

x = 1/16, y = 8/9


square9

x = 9/16, y = 1/27…


square_anim

animated square


Read full page: For Revver and Fevver

Pigmental Paradox

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:

William S. Burroughs
Alan Moore, C.B.E.
Michael Moorcock
Will Self
Stewart Home
Cormac McCarthy
Dr Joan Jay Jefferson
Serpent’s Tail
Titans of Trangression

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:

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):

p_pentomino_a


p_pentomino_b


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:

p_pentomino_a_anim


p_pentomino_a_anim


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:

p_pentomino_a_124_1

Pentomino-based fractal stage 1


p_pentomino_a_124_2

Pentomino-based fractal stage 2


p_pentomino_a_124_3

Pentomino-based fractal stage 3


p_pentomino_a_124_4

Stage 4


p_pentomino_a_124_5

Stage 5


p_pentomino_a_124_6

Stage 6


p_pentomino_a_124_7

Stage 7


p_pentomino_a_124_8

Stage 8


p_pentomino_a_124_9

Stage 9


p_pentomino_a_124_10

Stage 10


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

p_pentomino_b_234

p_pentomino_b_234anim

Animated version


p_pentomino_b_134

p_pentomino_b_134anim

Animated version


p_pentomino_b_124

p_pentomino_b_124anim


p_pentomino_b_123

p_pentomino_b_123anim


p_pentomino_a_134anim

p_pentomino_a_134


p_pentomino_a_234anim

p_pentomino_a_234


p_pentomino_a_124

p_pentomino_a_124anim


p_pentomino_a_123

p_pentomino_a_123anim


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:

p_pentomino_adj_13

p_pentomino_adj_anim13


p_pentomino_adj_6

p_pentomino_adj_anim6


p_pentomino_adj_anim5

p_pentomino_adj_5


p_pentomino_adj_3

p_pentomino_adj_anim3


p_pentomino_adj_2

p_pentomino_adj_anim2


p_pentomino_adj_1

p_pentomino_adj_anim1


p_pentomino_adj_17

p_pentomino_adj_anim17


p_pentomino_adj_15

p_pentomino_adj_anim15


p_pentomino_adj_16

p_pentomino_adj_anim16


p_pentomino_adj_8

p_pentomino_adj_anim8


p_pentomino_adj_10

p_pentomino_adj_anim10


p_pentomino_adj_11

p_pentomino_adj_anim11


p_pentomino_adj_14

p_pentomino_adj_anim14


p_pentomino_adj_anim4


p_pentomino_adj_anim12


p_pentomino_adj_anim9


p_pentomino_adj_anim7