Tutelary Trinity

“There are three golden rules to ensure computer security. They are: do not own a computer; do not power it on; do not use it.” — Robert H. Morris (1932-2011), computer scientist and once head of the NSA.

Can You Dij It? #1

The most powerful drug in the world is water. The second most powerful is language. But everyone’s on them, so nobody realizes how powerful they are. Well, you could stop drinking water. Then you’d soon realize its hold on the body and the brain.

But you can’t stop using language. Try it. No, the best way to realize the power of language is to learn a new one. Each is a feast with different flavours. New alphabets are good too. The Devanagari alphabet is one of the strongest, but if you want it in refined form, try the phonetic alphabet. It will transform the way you see the world. That’s because it will make you conscious of what you’re already subconsciously aware of.

But “language” is a bigger category that it used to be. Nowadays we have computer languages too. Learning one is another way of transforming the way you see the world. And like natural languages – French, Georgian, Tagalog – they come in different flavours. Pascal is not like Basic is not like C is not like Prolog. But all of them seem to put you in touch with some deeper aspect of reality. Computer languages are like mathemagick: a way to give commands to something immaterial and alter the world by the application of will.

That feeling is at its strongest when you program with machine code, the raw instructions used by the electronics of a computer. At its most fundamental, machine code is simply a series of binary numbers controlling how a computer processes other binary numbers. You can memorize and use those code-numbers, but it’s easier to use something like assembly language, which makes machine-code friendlier for human beings. But it still looks very odd to the uninitiated:

setupnum:
xor ax,ax
xor bp,bp
mov cx,20
clearloop:
mov [di+bp],ax
add bp,2
loop clearloop
ret

That’s almost at the binary bedrock. And machine code is fast. If a fast higher-level language like C feels like flying a Messerschmitt 262, which was a jet-plane, machine-code feels like flying a Messerschmitt 163, which was a rocket-plane. A very fast and very dangerous rocket-plane.

I’m not good at programming languages, least of all machine code, but they are fun to use, quite apart from the way they make you feel as though you’re in touch with a deeper aspect of reality. They do that because the world is mathematics at its most fundamental level, I think, and computer languages are a form of mathematics.

Their mathematical nature is disguised in a lot of what they’re used for, but I like to use them for recreational mathematics. Machine-code is useful when you need a lot of power and speed. For example, look at these digits:

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

They’re what the Online Encyclopedia of Integer Sequences (OEIS) calls “the almost natural numbers” (sequence A007376) and you generate them by writing the standard integers – 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13… – and then separating each digit with a comma: 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3… The commas give them some interesting twists. In a list of the standard integers, the 1st entry is 1, the 10th entry is 10, the 213rd entry is 213, the 987,009,381th entry is 987,009,381, and so on.

But that doesn’t work with the almost natural numbers. The 10th entry is 1, not 10, and the 11th entry is 0, not 11. But the 10th entry does begin the sequence (1, 0). I wondered whether that happened again. It does. The 63rd entry in the almost natural numbers begins the sequence (6, 3) – see the asterisks in the sequence above.

This happens again at the 3105th entry, which begins the sequence (3, 1, 0, 5). After that the gaps get bigger, which is where machine code comes in. An ordinary computer-language takes a long time to reach the 89,012,345,679th entry in the almost natural numbers. Machine code is much quicker, which is why I know that the 89,012,345,679th entry begins the sequence (8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 9):

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 63, 3105, 43108, 77781, 367573, 13859021, 77911127, 911360799, 35924813703, 74075186297, 89012345679…

And an ordinary computer-language might give you the impression that base 9 doesn’t have numbers like these (apart from the trivial 1, 2, 3, 4, 5, 6, 7, 8, 10…). But it does. 63 in base 10 is a low-hanging fruit: you could find it working by hand. In base 9, the fruit are much higher-hanging. But machine code plucks them with almost ridiculous ease:

1, 2, 3, 4, 5, 6, 7, 8, 10, 570086565, 655267526, 2615038272, 4581347024, 5307541865, 7273850617, 7801234568…

Poulet’s Propeller

The Penguin Dictionary of Curious and Interesting Numbers (1986) is one of my favourite books. It’s a fascinating mixture of math, mathecdote and math-joke:

2·618 0333…

The square of φ, the golden ratio, and the only positive number such that √n = n-1. (pg. 45)


6

Kepler discussed the 6-fold symmetry of snowflakes, and attempted to explain it by considering the close packing of spheres in a hexagonal array. (pg. 69)


39

This appears to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number to have the property of being uninteresting.

It is therefore also the first number to be simultaneously interesting and uninteresting. (pg. 120)

David Wells, who wrote the Dictionary, “had the rare distinction of being a Cambridge scholar in mathematics and failing his degree”. He must be the mathematical equivalent of the astronomer Patrick Moore: a popularizer responsible for opening many minds and inspiring many careers. He’s also written books on geometry and mathematical puzzles. But not everyone appreciates his efforts. This is a sideswipe in a review of William Hartston’s The Book of Numbers:

Thankfully, this book is more concerned with facts than mathematics. Anyone wanting to learn more about [π] or the Fibonacci sequence should turn to the Penguin Dictionary of Curious and Interesting Numbers, a volume which none but propeller-heads will find either curious or interesting. (Review in The Independent, 18th December 1997)


Continue reading: Poulet’s Propeller

Electrify Your Eyes

Front cover of The Spark of Life by Frances AshcroftThe Spark of Life: Electricity in the Human Body, Frances Ashcroft (Penguin 2013)

“Electricity in the Human Body” is the subtitle of this book. Make that the goat, frog, eel, shark, torpedo-ray, snake, platypus, spiny anteater, sooty shearwater and fruit-fly body too. And if Venus flytraps, maize and algae have bodies, throw them in next. Frances Ashcroft gives you a bargeload of buzz for your buck, a shedload of shock for your shekel: The Spark of Life describes the use of electricity by many different forms of life. But it discusses death a lot too, from lightning-strikes and electric chairs to heart-attacks and toxicology. Poisons can be a cheap and highly effective way of interfering with the electro-chemistry of the body:

The importance of sodium and potassium channels in generating the nerve impulse is demonstrated by the fact that a vast array of poisons from spiders, shellfish, sea anemones, frogs, snakes, scorpions and many other exotic creatures interact with these channels and thereby modify the function of nerve and muscle. … The tetrodotoxin contained in the liver and other tissues of this fish [the fugu or puffer-fish, Takifugu spp., Lagocephalus spp., etc] is a potent blocker of the sodium channels found in your nerves and skeletal muscles. It causes numbness and tingling of the lips and mouth within as little as thirty minutes … This sensation of “pins and needles” spreads rapidly to the face and neck, moves onto the fingers and toes, and is then followed by gradual paralysis of the skeletal muscles … Ultimately the respiratory muscles are paralysed, which can be fatal. The heart is not affected, as it has a different kind of sodium channel that is far less sensitive to tetrodotoxin. The toxin is also unable to cross the blood-brain barrier so that, rather horrifyingly, although unable to move and near death, the patient remains conscious. (ch. 3, “Acting on Impulse”, pp. 69-70)

In short, fugu-poisoning is the opposite of electrocution: it’s the absence rather than the excess of electricity that kills its victims. Those “channels” are a reminder that electro-chemistry could also be called electro-mechanics: unlike an electricity-filled computer, an electricity-filled body has moving parts – and in more ways than one. Our muscles move because ions move in and out of our cells. This means that a body has to be wet inside, not dry like a computer, but it’s easy to imagine a human brain controlling a robotic body. But would a brain still be conscious if it became metal-and-plastic too? Perhaps a brain has to be both soggy and sparky to be conscious.

The electrical nature of the brain certainly seems important, though that may be a superstitious conclusion. Electricity is a mysterious phenomenon and so is consciousness, so they seem to go together well. Ashcroft writes a lot about the sense-organs and the data they supply to the brain, but like all scientists she cannot explain how those data are turned into conscious experience as the maths-engine of the brain applies its neuro-functions and neuro-algorithms. However, she does suggest ways in which our consciousness might be expanded in future. Humans have colour vision, based on the three types of cone-cells in our eyes:

Most mammals, such as cats and dogs, have only two types of cone photopigment and so see only a limited range of colour … Other animals live in a world entirely without colour. But humans should not be too complacent, for we are far from having the best colour vision in the animal world and lag far behind the mantis shrimp, which enjoys ten or more different visual pigments. Even tropical fish possess four or five types of cones. (ch. 9, “The Doors of Perception”, pg. 199)

Bio-engineering may one day sharpen and extend all our senses, from sight and hearing to touch, taste and smell. It may also give us new senses, like the ability to form sound-pictures like bats and detect infra-red like pit-vipers. And why not X-rays and radio-waves too? It’s an exciting prospect, but in a sense it won’t be anything new: our new senses, like our old ones, will depend on nerve-impulses and the way they’re mashed and mathed in that handful of “electrified clay” known as the brain.

“Electrified clay” is Shelley’s phrase: like his wife Mary, he was fascinated by the early electric experiments of the Italian scientists Luigi Galvani and Alessandro Volta. Mary turned her fascination into a book called Frankenstein (1818) and her invention is part of the scientific history in this book. The story of bio-electricity is still going strong: there are electric mysteries in all kinds of bodies waiting to be solved. Maybe consciousness is one of them. And if science proves unable to crack consciousness, it’s certainly able to expand it. Reading this book is one way to experience the mind-expanding powers of science, but seeing like a mantis shrimp would be good too.