# 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 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 → 1/2
2: halton(10) = 01/100 → 1/4
3: halton(11) = 11/100 → 3/4
4: halton(100) = 001/1000 → 1/8
5: halton(101) = 101/1000 → 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 → 1/3
2: halton(2) = 2/10 → 2/3
3: halton(10) = 01/100 → 1/9
4: halton(11) = 11/100 → 4/9
5: halton(12) = 21/100 → 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?