PLAYING THE NUMBERS

Dick Pountain/ Idealog 291/ 6th October 2018 13:41:14

Most people encounter turning points in their lives, and one of the earliest in mine came while deciding to apply for university: I loved both chemistry and maths, and had the A level passes to study either, so I could quite easily have become a mathematician, but at that time the lure of smells and bangs was just a bit stronger. However I’ve never lost my love of mathematics, especially number theory, and still do some for fun from time to time. Back in the sixties I spent months playing at topology, in particular notations for classifying knots. More recently – as I’ve mentioned here before – I wanted to hear what the prime numbers sound like, which lead me into a whole computer music composition project.

A subject of particular fascination for me has always been irrational numbers. If you remember your school maths, these are numbers that cannot be expressed as fractions – that is as the ratio of two whole numbers like ½ or 3793/8331. Examples are pi and , which can only be represented by infinitely-long, non-repeating strings of decimal (or binary, hex, octal etc) digits, which are of course impossible to complete. And the weirdest thing about irrational numbers is how many of them there are....

You can present all the numbers as a line that stretches off forever to left and right, on which every point represents a different number. You can tick off the whole numbers along this line, like the marks on a ruler, and in between each mark there will be infinitely many rational numbers, fractions whose decimal representation comes to an end like 4.567757, and infinitely many irrational numbers. The scary thing is that there are ‘more’ (whatever that means) irrationals than anything else, because if you choose a point at random on the line, there’s a 100% chance it will be irrational! I know, it made me feel seasick too when I learned this, from an article in Quanta Magazine called ‘Why Mathematicians Can’t Find the Hay in a Haystack’ (https://www.quantamagazine.org/why-mathematicians-cant-find-the-hay-in-a-haystack-20180917). Almost all the numbers we use in everyday life are rational, whole numbers like ‘12 eggs’ or fractions like £4.56, but these are like needles in an infinite haystack of irrationals, impossible to locate by random search. And worse still, the hay itself is impossible to describe in finite space.

Rational numbers are easier to handle, and of course they’re all we can deal with on a computer that has both finite memory and finite word-length. But even rational numbers can boggle the mind. Another of my passions is computer graphics, and images on a computer screen are really nothing but tables of numbers, displayed as colour intensities by the graphics hardware. Suppose your computer has a 1920 x 1080 HD display and 32-bit colour, then every line on that screen could be represented by a 61,440-bit long binary number, and every possible image by a 66,355,200-bit number (61440*1080).

So, you could write a program that counts from 0 to 66,355,200, converts each number to a bitmap and displays it, and it would eventually show everything that could be possibly seen on that screen, like those proverbial monkeys on typewriters that might produce Shakespeare. It wouldn’t take infinite time either: if you showed each picture for a second you’d be done in two years, though I wouldn’t sit up waiting for the Mona Lisa to appear.

Actually the more I think about this, the more I want to set it up and present it to some gallery as an art project - people might sit in front of it hoping to see something recognisable, like the librarians in Jorge Luis Borges’ Library of Babel searching for a legible word.

Maths, when properly appreciated, is like a good roller coaster that induces vertigo, nausea and pleasure all at the same time. For me, the ultimate boggle is Cantor’s ‘diagonal’ proof that there are infinitely more real numbers, rational and irrational, than there are whole numbers. Build a table, infinitely deep and infinitely wide, of all the real numbers (which will take a while). Traverse this table diagonally, taking first digit of line one, second digit of line two and so on, add one to it and string these all together into a new infinitely-long number. This new number can’t be anywhere in the table because it differs from every number in the table in one place, so infinity squared isn’t big enough to hold them all – they are uncountable. First encountered at school, this still hurts my head a little even now. Perhaps I made the right choice after all: in chemistry you can only poison yourself or blow yourself up.  

FUTURE IMPERFECT

Dick Pountain/ Idealog 290/ 8th September 2018 08:44:43

For the sake of my health I walk on beautiful Hampstead Heath at least once a week, and I get there via the revamped London Overground train service. It’s clean, frequent and gets me there in less than 15 minutes. “So what?” you may be thinking, but the significance is I get there by train rather than by flying car.

“Where’s my flying car?” is one of those amusing/stupid tropes one sees pretty regularly online, and like many such tropes it contains a kernel of important truth. We don’t have flying cars yet, and almost certainly never will, and the reason has little to do with technology. YouTube is crammed with videos of workable flying cars, many of them good-looking, all of them too expensive (but of course that price could crash if they ever went into mass production). The reason they don’t go into mass production is of course social, which means also political. We live in a complex industrial society in which many resources have to be rationed and shared, urban space being one of them. On paper putting everyone into a flying car ought to economise on that space, since the ordinary motor car (or bus or train) needs roads laid out on an almost-flat 2D surface.

Flying cars would bring the third dimension into use - except that the third dimension is already in use by airlines. In mass use it would be impossible to police (traffic lights?) Fixed-wing or drone, a flying car is still subject to gravity and if two collide they’re likely end up falling through someone’s roof. The problems are endless and flying cars could only ever be tolerable in small numbers. Of course they already are - they’re called helicopters, and mostly used by soldiers and billionaires. The rest of us must share flying trains called airliners.

My point is that ‘futurists’ often get things hilariously wrong, not because they’re stupid, but because their expertise and experience is narrow: they may be brilliant in one technical field but nerdishly disconnected from normal society and shamefully ignorant of its basic requirements. A recent BBC 4 retrospective covered fifty years of Horizon programmes, in which revered sci-fi authors Arthur C Clarke and Isaac Asimov made predictions about AI and robotics that now look quaintly naive, while Ray Kurzweil’s ‘grey goo’ warning looked frankly unhinged (even if Prince Charles did believe it…). The tendency to overestimate the effect of new technologies is not itself new.

One futurist that I do respect is Douglas Rushkoff, who just wrote an inflammatory piece with the snappy title “How tech's richest plan to save themselves after the apocalypse” (on Medium, but reprinted by the Guardian https://www.theguardian.com/technology/2018/jul/23/tech-industry-wealth-futurism-transhumanism-singularity) He was recently paid half his annual academic salary by five hedge-fund managers to talk about future technology, but it turned out they weren’t really interested in tech per se, but in the coming apocalypse it promised. They realised that armed guards would be required to protect their New Zealand or Alaskan bunkers from the angry mobs "but how would they pay the guards once money was worthless? What would stop the guards from choosing their own leader? The billionaires considered using special combination locks on the food supply that only they knew”.

Enormous riches tempt people to believe their possessor must be very clever, but clever at what? Clever at designing social networks no more qualifies you to redesign society than being clever at property speculation qualifies you to be President. Enormous riches frequently accumulated through what economists call ‘network effects’: certain inventions, like railways, telephones, gramophone records, television, took off exponentially only once enough people had the means to use them, so as much through luck as technical superiority. But riches tempted their possessors to believe themselves invincible, and high-tech entrepreneurs are particularly prone to drawing their notions of invincibility from sci-fi, comics and video games. As Rushkoff put it, they consider human evolution to be “a video game that someone wins by finding the escape hatch and then letting a few of his BFFs come along for the ride. Will it be Musk, Bezos, Thiel … Zuckerberg?”

Transport For London (TfL), which regulates the capital’s bus, underground, overground and taxi services using a mixture of public and private suppliers, is a dull thing indeed compared to Elon Musk’s plan to fire us, inside tin cans, at supersonic speed though vacuum tubes, but it satisfactorily moves a lot of people each day using little in the way of sci-fi magic except for the Oyster card system (which renders transport cashless so effectively that we no longer notice). I’d feel far happier if TfL were running the world than Rushkoff’s humanicidal billionaires...

BEHIND THE CURVE

Dick Pountain/ Idealog289/ 7th August 2018 09:44:18

Regular readers might know that a favourite pastime of mine is creating computer art. For the last 12 years or so this consisted mainly of taking photographs, some of which I posted to Flickr straight but other times I’d modify to look like paintings, using my preferred tool Photoshop Elements. These modifications depended heavily on Photoshop’s layers, which I’d duplicate, filter and re-merge over and again until the visual result pleased me. My discovery of Silver Efex Pro – one of the Nik filter suite Google that so magnanimously gave away – was a leap forward as its ‘structure’ parameter gives great control over the levels of detail in a picture.

But regular readers may also know that over a year ago I adopted a Chromebook as my main computer, and Chromebooks don’t run Photoshop (except for Express which won’t do). I tried many photoeditor offerings like Photosuite and Pixlr, which have layers, but the only one that clicked was Sumo Paint, which looks a bit like PS Elements, and has some crazy fills, distortions and filters that at first I found a bit OTP. But one fine day I tried its Fractal Morpher filter and found a new hobby.

All Fractal Morpher does is take a simple line drawing and project it into a complex fractal figure around symmetry axes you can vary from 1 to 16, and I found the patterns it produces fascinating. They’re not the shiny, 3D fractal objects and scenes of so much online fractal art, but are strictly 2D – what grabbed me about them is their gorgeous curvature.

It’s not surprising that a fractal algorithm should produce curves that are visually attractive, because so many curves in nature are produced by similar ‘algorithms’ executed by animals’ and plants’ metabolisms. That sunflower seeds are arranged in spirals described by the Fibonacci Series is old news. D’Arcy Thompson described all this 100 years ago in ‘On Growth and Form’; Alan Turing did some of the maths, as did the Russian Ilya Prigogine; in 1990 Prusinkiewicz and Lindenmayer, in ‘The Algorithmic Beauty of Plants’ even extracted some of these algorithms and coded them in C++ (see my Idealog 234, 7th Jan 2014).

Since we’re animals too, why wouldn’t our brains prefer some curves over others? Modern artists I love like Picasso, Miro, Klee, Kandinsky, Malevich all unfailingly draw curves that are ‘right’, in the sense that they could once have been alive, while many artists I like less draw ‘bad’ curves that couldn’t. Fractal Morpher always produces curves that are right.

I started making crude but highly coloured line drawings, feeding them through Fractal Morpher and then applying my layering techniques to produce abstract ‘paintings’ so psychedelically detailed that they look like they might have taken months to do, but in fact took minutes. It actually became habit-forming: if I ever feel at a loose end, it’s very satisfying to dash off another. (I can’t show you one here, but there’s a fair selection on Facebook at

https://www.facebook.com/media/set/?set=a.10157601291269478.1073741843.636729477&type=1&l=c4a07f83f0 )

I deliberately used the word psychedelic back there because these images, as well as seeming ‘natural’, display a deep complexity that resembles the hallucinations produced by LSD. (And yes I did, back in the 1960s, and no, I don’t nowadays). This too is hardly surprising because LSD works on the human perceptual system which is made up of neurons - in the eye’s retina and the visual cortex - that grow and connect according to rules similar to those that govern sunflower seeds or snail shells. There’s a good article by Jennifer Ouellette in Quanta Magazine (https://www.quantamagazine.org/a-math-theory-for-why-people-hallucinate-20180730/) about the mathematics of Acid hallucinations. Basic shapes like lattices, cobwebs, honeycombs, spirals and tunnels recur because they do indeed reflect connection patterns in the retina and visual cortex, where they are components of the mechanisms via which we dissect and analyse perceived scenes. Our brain doesn’t employ Cartesian coordinates like a computer screen but these more organic forms.

So I must now be entirely happy with my graphical facilities on Chromebook right? Actually I wasn’t quite, because Sumo Paint lacked several of the Blend Modes available in PS that I used a lot, and also the Nik filters. Then came that upgrade to ChromeOS which let me run Android Apps, and that enabled me to use the marvellous Snapseed: a photoeditor with a unique and intuitive UI designed for finger use on phones and tablets. And, oh joy, several of its built-in filters replicate the actions of Silver Efex Pro, in particular that all-important Structure parameter. There’s now almost nothing that I need to go back to a Windows machine to do, which feels profoundly liberating.