Self-Raising Power

The square root of 2 is the number that, raised to the power of 2, equals 2. That is, if r^2 = r * r = 2, then r = √2. The cube root of 2 is the number that, raised to the power of 3, equals 2. That is, if r^3 = r * r * r = 2, then r = [3]√2.

But what do you call the number that, raised to the power of itself, equals 2? I suggest “the auto-root of 2”. Here, if r^r = 2, then r = [r]√2. I don’t know a quick way to calculate the auto-root, but you can adapt a well-known algorithm for approximating the square root of a number. The square-root algorithm looks like this:

n = 2
r = 1
for c = 1 to 20
    r = (r + n/r) / 2
next c
print r

r = 1.414213562…

Note the fourth line of the algorithm: r = (r + n/r) / 2. When r is an over-estimate of √2, then 2/r will be an under-estimate (and vice versa). (r + 2/r) / 2 splits the difference and refines the estimate. Using the lines above as the model, the auto-root algorithm looks like this:

n = 2
r = 1
for c = 1 to 20
    r = (r + [r]√n) / 2[*]
next c
print r

r = 1.559610469…


*This is equivalent to r = (r + n^(1/r)) / 2

Here are the first 100 digits of [r]√2 = r in base 10:

1, 5, 5, 9, 6, 1, 0, 4, 6, 9, 4, 6, 2, 3, 6, 9, 3, 4, 9, 9, 7, 0, 3, 8, 8, 7, 6, 8, 7, 6, 5, 0, 0, 2, 9, 9, 3, 2, 8, 4, 8, 8, 3, 5, 1, 1, 8, 4, 3, 0, 9, 1, 4, 2, 4, 7, 1, 9, 5, 9, 4, 5, 6, 9, 4, 1, 3, 9, 7, 3, 0, 3, 4, 5, 4, 9, 5, 9, 0, 5, 8, 7, 1, 0, 5, 4, 1, 3, 4, 4, 4, 6, 9, 1, 2, 8, 3, 9, 7, 3…

And here is [r]n = r for n = 2..20:

autopower(2) = 1.5596104694623693499703887…
autopower(3) = 1.8254550229248300400414692…
autopower(4) = 2
autopower(5) = 2.1293724827601566963803119…
autopower(6) = 2.2318286244090093673920215…
autopower(7) = 2.3164549587856123013255030…
autopower(8) = 2.3884234844993385564187215…
autopower(9) = 2.4509539280155796306228059…
autopower(10) = 2.5061841455887692562929409…
autopower(11) = 2.5556046121008206152514542…
autopower(12) = 2.6002950000539155877172082…
autopower(13) = 2.6410619164843958084118390…
autopower(14) = 2.6785234858912995813011990…
autopower(15) = 2.7131636040042392095764012…
autopower(16) = 2.7453680235674634847098492…
autopower(17) = 2.7754491049442334313328329…
autopower(18) = 2.8036632456580215496843618…
autopower(19) = 2.8302234384970308956026277…
autopower(20) = 2.8553085030012414128332189…

I assume that the auto-root is always an irrational number, except when n is a perfect power of suitable form, i.e. n = p^p for some integer p. For example, autoroot(4) = 2, because 2^2 = 4, autoroot(27) = 3, because 3^3 = 27, and so on.

And here is the graph of autoroot(n) for n = 2..10000:
autoroot

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Performativizing Papyrocentricity #29

Papyrocentric Performativity Presents:

Sky StoryThe Cloud Book: How to Understand the Skies, Richard Hamblyn (David & Charles 2008)

Wine WordsThe Oxford Companion to Wine, ed. Janice Robinson (Oxford University Press 2006)

Nu WorldsNumericon, Marianne Freiberger and Rachel Thomas (Quercus Editions 2014)

ThalassobiblionOcean: The Definitive Visual Guide, introduction by Fabien Cousteau (Dorling Kindersley 2014) (posted @ Overlord of the Über-Feral)


Or Read a Review at Random: RaRaR

Breeding Bunnies

Front cover of The Golden Ratio by Mario Livio
The Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty, Mario Livio (Headline Review 2003)

A good short popular guide to perhaps the most interesting, and certainly the most irrational, of all numbers: the golden ratio or phi (φ), which is approximately equal to 1·6180339887498948482… Prominent in mathematics since at least the ancient Greeks and Euclid, phi is found in many places in nature too, from pineapples and sunflowers to the flight of hawks. Livio catalogues its appearances in both maths and nature, looking closely at the Fibonacci sequence and rabbit-breeding, before going on to debunk mistaken claims that phi also appears a lot in art, music and poetry. Dalí certainly used it, but da Vinci, Debussy and Virgil almost certainly didn’t. Nor, almost certainly, did the builders of the Parthenon and pyramids. Finally, he examines what has famously been called (by the physicist Eugene Wiegner) the unreasonable effectiveness of mathematics: why is this human invention so good at describing the behaviour of the Universe? Livio quotes one of the best short answers I’ve seen:

Human logic was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. That is why mathematics is consistent with the physical world. (ch. 9, “Is God a mathematician?”, pg. 252)

It’s not hard to recommend a book that quotes everyone from Johannes Kepler and William Blake to Lewis Carroll, Christopher Marlowe and Jef Raskin, “the creator of the Macintosh computer”, whose answer is given above. Recreational mathematicians should also find lots of ideas for further investigation, from fractal strings to the fascinating number patterns governed by Benford’s law. It isn’t just human beings who look after number one: as a leading figure, 1 turns up much more often in data from the real world, and in mathematical constructs like the Fibonacci sequence, than intuition would lead you to expect. If you’d like to learn more about that and about many other aspects of mathematics, hunt down a copy of this book.


Elsewhere other-posted:

Roses Are Golden – φ and floral homicide

The World as Worm

In “Hymn to Herm”, I wrote about a religion based on √2, or the square root of two, the number that, multiplied by itself, equals 2. In the religion, neophytes learn the mystery and majesty of this momentous number when they try to calculate its exact value. The calculation involves adding and subtracting fractions based on powers of two. The first step is this: 1 x 1 = 1. So that’s too small. Add 1/2^1 = ½ and re-multiply: 1½ x 1½ = 2¼. Too big. So subtract 1/2^2 = ¼, and re-multiply. 1¼ x 1¼ = 1+9/16. Too small. Add 1/8 and re-multiply. 1+3/8 x 1+3/8 = 1+57/64. Too small again. Add 1/16 and re-multiply. And so on.

In effect, what the neophytes are doing is calculate the digits of √2 in binary, or base two. When the multiplication is too small, put a 1; when it’s too big, put a 0. Like this:

1 x 1 = 1 < 2, so √2 ≈ 1·…
1½ x 1½ = 2¼ > 2, so √2 ≈ 1·0…
1¼ x 1¼ = 1+9/16 < 2, so √2 ≈ 1·01…
(1+3/8) x (1+3/8) = 1+57/64 < 2, so √2 ≈ 1·011…
(1+7/16) x (1+7/16) = 2+17/256 > 2, so √2 ≈ 1·0110…
(1+13/32) x (1+13/32) = 1+1001/1024 < 2, so √2 ≈ 1.01101…
(1+27/64) x (1+27/64) = 2+89/4096 > 2, so √2 ≈ 1.011010…
(1+53/128) x (1+53/128) = 1+16377/16384 < 2, so √2 ≈ 1·0110101…
(1+107/256) x (1+107/256) = 2+697/65536 > 2, so √2 ≈ 1·01101010…
(1+213/512) x (1+213/512) = 2+1337/262144 > 2, so √2 ≈ 1·011010100…
(1+425/1024) x (1+425/1024) = 2+2449/1048576 > 2, so √2 ≈ 1·0110101000…
(1+849/2048) x (1+849/2048) = 2+4001/4194304 > 2, so √2 ≈ 1·01101010000…
(1+1697/4096) x (1+1697/4096) = 2+4417/16777216 > 2, so √2 ≈ 1·011010100000…
(1+3393/8192) x (1+3393/8192) = 1+67103361/67108864 < 2, so √2 ≈ 1·0110101000001…

Mathematically naïve neophytes, seeing the process miss 2 by smaller and smaller amounts on either side, might imagine that eventually the exact root will appear and the calculations end. But they would be wrong. They could work a year or a million years: they would never calculate the exact square root of two. There is no ratio of whole numbers, a/b, such that a^2/b^2 = 2. In other words, √2 is an irrational number, or number that can’t be represented as a ratio of integers (please see appendix for the proof).

This discovery, made by Greek mathematicians more than two millennia ago, is both mind-boggling and world-shattering. In fact, it’s mind-boggling in part because it’s world-shattering. √2 shatters the world because the world is too small to contain it: in the words of the Cult of Infinite Hermaphrodites, “Were the sky all parchment, the seas all ink, and gulls all plucked for quills”, the square root of two could not be recorded in full. This is far more certain than tomorrow’s sunrise, because predicting tomorrow’s sunrise depends on fallible scientific reasoning from incomplete knowledge. Proving the irrationality of √2 depends on infallible mathematical reasoning.

At least, it’s as close to infallible as human beings can get. But that’s another part of what is mind-boggling about √2. A finite, feeble human being, with a speck of soon-decaying brain, can prove the existence of things larger than the universe. A few binary digits of √2 are shown above. Here are a few more:

1·
0110101000001001111001100110011111110011101111001100100100001000
1011001011111011000100110110011011101010100101010111110100111110
0011101011011110110000010111010100010010011101110101000010011001
1101101000101111010110010000101100000110011001110011001000101010
1001010111111001000001100000100001110101011100010100010110000111
0101000101100011111111001101111110111001000001111011011001110010 
0001111011101001010100001011110010000111001110001111011010010100 
1111000000001001000011100110110001111011111101000100111011010001 
1010010001000000010111010000111010000101010111100011111010011100 
1010011000001011001110001100000000100011011110000110011011110111 
1001010101100011011110010010001000101101000100001000101100010100 
1000110000010101011110001110010001011110111110001001110001100111 
1000110110101011010100010100011100010111011011111101001110111001 
1001011001010100110001101000011001100011111001111001000010011011 
1110101001011110001001000001111100000110110111001011000001011101 
1101010101001001010000010001001100100000100000011001010010010101 
0000001001110010100101010110110110110001111110100001110111111011 
1110100110100111010000000101100111010111100100100111110000011000 
1000010011001001101101010111100110101010010100010110110010100011 
0111000110011110011010000011011011011111000001000110110110001110 
0000001000001001101110000000001111111100011001000110101001011110 
0110011001010100101111010011111011110111101101000011110101111111 
1110110101000011011111000111111110010100010001000010011000001111 
1011110101000000110001001000001111101111010101010000001110000101 
1000001111111001011110111011110101000101111011111011100001100110 
0011000100000111000101000101110101011111111010111110011101100101 
1010010010011110100101001110110001111111010110010111000100000101 
1111101111111100001011100001111110100111011000111110111100000001 
1111001101011001100111001000001011110010111111100101000000001011 
1000010010001100111100001011110100100101001010101110000001000110 
1011111110011111000111101111011110010100011111010100011001110110 
1001101011111000110000010100101111001100011001111100011111000010 
1001000010111110011101101001001010011011000001010111100011000001 
0000101101011000010011111011010010000111110010010010010011110101 
1011011100011111100000101101110011010010100100000011011000001001 
1101111011101000100100010010100110000011110101001110101010101101 
0000111011101010001100100001111101110100100010011111010001101010 
0111111010010000001100001011111000100000111110110111011010010100 
1110111110110101100011001001100110000100110011011101011100001010 
0001110110101001000001000101110000111101000100110011101000000110 
1000010000100011110101101110001110000011000000111101100100000001 
1011101010011101101000110100011101100110100001000111100101101100 
0101110011010101100101110010110111000000111111110011010101000000 
1100001101000001001010010100001011010110010000000110000100000001 
1110111101101111110001101101111010010001000101001010001010110100 
1111001001001000110001101000100111000110000000001011101101000000 
1010100010110101011010110000010000011111110101011101111001101110 
0000110111010000110001100110110101001000001100011111111001111111 
1111111101010111010101111110010001110001000010011000000011001101 
1011110101011100001001101000010010000101110110100101111010010001 
1011001111100010111100100000010110110111001001110010010110111001 
0111000111010110000010100001111110001000100011110000100010100000 
1010011011100001000000001100110011101101110000101100111001011011 
1101100110001010111011100111000111100100001011100010011010001101 
0011011110100110000001110010111100100010000000100011010001100001 
0011111111111100001000100100010100110100001110011110101010010111 
1010100110011001101101101100100111100011110011100111000111111001 
0100110101100000100100101010110011100001001000001010101110001110 
0101010100001110000011010101010100010001011010001000011000110001 
0111011110001100111101100000001101010000110100000010111111101000 
0101111100101001111011001000101111100101110001110010101110000000 
0111101011110101011101110001101110000010010110110011000010100000 
1110011110000011011101101010100100011100000010001100011010100111 
1111000011111000111100110010001110110011011000101000000111010010 
0010010101101000100111000000101101011010100000100000010001111101 
1011100110001001111101100011101010001010011001001110100001010001 
1001101111000000110100001100000111100010001000101000000001001000 
0100110110010100111101001111100110111011001111010100101100110001 
1101010010001001110101110101001000110001101101011100011000110011 
1100010010000000110010010110101111100101010010011011111101011101 
1001011001100111100010110100110100101100010011011101101010000110 
0111101111011000111001001000000000101001111111101010100011001000 
0001011100110101011001111100001010111010001111010011110011101001 
1101111111100000101111010001101101001101110101110111000100010111 
1000000001010111101101101001010110110111111010101111000110110000 
0101110000100010110010001101010111111110111010111101000001110111 
1111111011001001011011011011100011110111011110001111110000011100 
0010101110111011110011100001101101001001111010111111010110101111 
0100010001100000100010000010100101011000101011011101000000011100 
1010011111110001101101101011110000001011011111101100000110111100 
0110111000001010011011101101101111000110011111111000010110110010 
0111010011100000100001100001101100111010000100110111010101110001 
1011000101100101010010011000011100111101001000010001110001101010 
1010111001101001110110000000000111100101011110010100010001011011 
1100011000001010001111100000101001001111110110001001011010001110 
1011011110010100101111011100011100000010110101101001010001011101 
1010100101001011000001001010010001000000110010101011110010010100 
0011100001111100001111010010011011111101000011110011101111101000 
1010111101100011000001011010010100111010000101110111001010001000 
1010110010100001001111111011010000000110110010011000001010010001 
0101110110000011101110100000110100110101010110001101100000011101 
1101000100010101100111101001011001000011111010101010001001111110 
1011011101110101011110100010000001010010100101110101101101101111 
0100101010001000100111100011110100001001001010111011000111000110 
1000010101001000000011011100001011101001100110010100011110110011 
0111001011011110110110100000010111100010000110010010111110010101 
1011000110111001001001100101000100101011010000000100110000110011
0001100000011101011...

The distribution of 1’s and 0’s seems effectively random, as though the God of Mathematics were endlessly tossing a coin, putting 1 for heads, 0 for tails. Yet √2 is the opposite of a random number. Change a single digit anywhere and it ceases to be √2. Every 1 and every 0 is rigidly determined by “unalterable law”. So are the position and magnitude of the digits of √2 in every other base. Here, for example, is √2 in base 4:

1·
112220021321212133303233030210020230233230103121232222111133
103320322313230011311010213131100212131220233112100230012121
303020222211133210012002013111...

Another word for base-4 is DNA: genes are in fact written in a base-4 code based on the chemicals guanine, adenine, thymine and cytosine, or G, A, T, C for short. If the digits of √2 are truly random, in the statistical sense, then all genomes, actual and potential, occur somewhere along its length: yours, mine, the Emperor Heliogabalus’s, Bilbo Baggins’, the sabre-toothed tiger’s, the dodo’s, and so on. But almost all the “DNA” of √2 in base-4 will be meaningless: although √2 is the opposite of random, it is effectively a typing chimpanzee. Or a typing worm – a type-worm. √2 is like an endless worm that types out its own segments on a typewriter with two keys (for binary numbers) or four keys (for quaternary numbers) or ten keys (for decimal numbers) and so on.

But √2 doesn’t just encode the genomes of individual people, animals and plants: it encodes everything they do throughout their lives. In fact, it encodes the entire universe. And perhaps the universe is √2 or some number like it. Perhaps, in some sense, everything exists within the digits of an irrational number, or a sufficiently large rational number. If so, then √2 has become aware of itself through human beings: the World as Worm has bitten its own tail.

Appendix: Proof of the irrationality of √2

1. Suppose that there is some ratio, a/b, such that

2. a and b have no factors in common and

3. a^2/b^2 = 2.

4. It follows that a^2 = 2b^2.

5. Therefore a is even and there is some number, c, such that 2c = a.

6. Substituting c in #4, we derive (2c)^2 = 4c^2 = 2b^2.

7. Therefore 2c^2 = b^2 and b is also even.

8. But #7 contradicts #2 and the supposition that a and b have no factors in common.

9. Therefore, by reductio ad absurdum, there is no ratio, a/b, such that a^2/b^2 = 2. Q.E.D.