Day 308: A Brief History Of Infinity

The infinite is a concept so remarkable, so strange, that contemplating it has driven at least two great mathematicians over the edge into insanity.

In the Hitch-hiker’s Guide to the Galaxy, Douglas Adams described how the writers of his imaginary guidebook got carried away in devising its introduction:

‘Space,’ it says, ‘is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the street to the chemist, but that’s just peanuts to space. Listen ...’ and so on. After a while the style settles down a bit and it starts telling you things you actually need to know ...

Infinity makes space seem small.

Yet this apparently unmanageable concept is also with us every day. My daughters were no older than six when they first began to count quicker and quicker, ending with a blur of words and a triumphant cry of ‘infinity!’ And though infinity may in truth make space seem small, when we try to think of something as vast as the universe, infinite is about the best label our minds can apply.

Anyone who has broken through the bounds of basic mathematics will have found the little ∞ symbol creeping into their work (though we will discover that this drunken number eight that has fallen into the gutter is not the real infinity, but a ghostly impostor). Physicists, with a carelessness that would make any mathematician wince, are cavalier with the concept. When I was studying physics in my last years at school, a common saying was ‘the toast rack is at infinity’. This referred to a nearby building, part of Manchester Catering College, built in the shape of a giant toast rack. (The resemblance is intentional, a rare example of humour in architecture. The companion building across the road, when seen from the air, looks like a fried egg.) We used the bricks on this imaginative structure to focus optical instruments. What we really meant by infinity was that the building was ‘far enough away to pretend that it is infinitely distant’.

Infinity fascinates because it gives us the opportunity to think beyond our everyday concerns, beyond everything to something more – as a subject it is quite literally mind-stretching. As soon as infinity enters the stage, it seems as if common sense leaves. Here is a quantity that turns arithmetic on its head, making it seem entirely feasible that 1 = 0. Here is a quantity that enables us to cram as many extra guests as we like into an already full hotel. Most bizarrely of all, it is quite easy to show that there must be something that is bigger than infinity – which surely should be the biggest thing there could possibly be.

Although there is no science more abstract than mathematics, when it comes to infinity, it has proved hard to keep spiritual considerations out of the equation. When human beings contemplate the infinite, it is almost impossible to avoid things theological, whether in an attempt to disprove or prove the existence of something more, something greater than the physical universe. Infinity has this strange ability to be many things at once. It is both practical and mysterious. Scientists and engineers use it quite happily because it works – but they consider it a black box, having the same relationship with it that most of us do with a computer or a mobile phone, something that does the job even though we don’t quite understand how.

The position of mathematicians is rather different. For them, modern considerations of infinity shake up the comfortable, traditional world in the same way that physicists suffered after quantum mechanics shattered the neat, classical view of the way the world operated. Reluctant scientists have found themselves having to handle such concepts as particles travelling backwards in time, or being in two opposite states at the same time. As human beings, they don’t understand why things should be like this, but as scientists they know that if they accept the picture it helps predict what actually happens. As the great twentieth-century physicist Richard Feynman said in a lecture to a non-technical audience:

It is my task to convince you not to turn away because you don’t understand it. You see, my physics students don’t understand it either. That is because I don’t understand it. Nobody does.

Infinity provides a similar tantalizing mix of the normal and the counter-intuitive.

All of this makes infinity a fascinating, elusive topic. It can be like a deer, spotted in the depths of a thick wood. You will catch a glimpse of beauty that stops you in your tracks, but moments later you are not sure if you saw anything at all. Then, quite unexpectedly, the magnificent animal stalks out into full view for a few, fleeting seconds.

A real problem with infinity has always been getting through the dense undergrowth of symbols and jargon that mathematicians throw up. The jargon is there for a very good reason. It’s not practical to handle the subject without some use of these near-magical incantations. But it is very possible to make them transparent enough that they don’t get in the way. We may then open up clear views on this most remarkable of mathematical creatures – a concept that goes far beyond sheer numbers, forcing us to question our understanding of reality.

Welcome to the world of infinity.

~~A Brief History Of Infinity : The Quest to Think the Unthinkable -by- Brian Clegg

Day 238: Spirals In Time

On the banks of the Kinta River, at the furthest navigable point inland from Peninsular Malaysia’s western coast, stands the former mining town of Ipoh. Behind the bustling Chinese shophouses, white colonial town hall and railway station lies a backdrop of some seventy limestone hills, clad in forest. As visitors climb steps to the Buddhist temples perched in these green humps, or descend into the caves beneath them, they are surrounded by biological treasures, including some of the world’s smallest and strangest shells.

Karst limestone formations, like the ones in Ipoh, can be seen throughout South-east Asia, from northern Vietnam through Cambodia and Thailand to the Philippines and Indonesia; they rise from the sea as idyllic islands, and poke through rainforest canopies. The limestones were formed millions of years ago by the remains of ancient sea creatures, including corals and shells. Since then, their calcium carbonate skeletons have been uplifted, then eroded by wind and rain into jagged silhouettes with giant caves inside them and underground rivers running through them.

A riot of unusual wildlife lives in these limestone landscapes. Bumblebee Bats, the world’s smallest mammals, flit through the caves; blind fish crawl from subterranean ponds and out onto rocks; beetles and millipedes prosper in huge piles of bat dung; and out on the rugged hilltops roam troops of leaf monkeys, including such incredibly rare species as the Delacour’s Langur with its striking black and white fur (the Vietnamese name for it, vooc mong trang, means ‘the langur with white trousers’). The chalky soils are also a haven for molluscs that find a plentiful supply of the principal raw material to make their shells.

A single Malaysian limestone hill can be home to between 40 and 60 species of tiny microsnails, each one a millimetre tall, and all of them with highly ornate shells. Of those, two or three species could be unique to that individual hill. As well as snails, there are heaps of other endemic species here, ones that are found nowhere else on the planet: geckos, crickets, orchids, begonias and spiders. Just like oceanic islands, the limestone outcrops are isolated dots of habitat where evolution dances to a different beat, generating new and peculiar species.

When biologist Reuben Clements went snail-hunting in the hills of Ipoh, he discovered a shell like no other. To find it, all he had to do was take a few scoops of soil, place them in a bucket of water and wait for the empty shells to rise to the surface (for a long time only recently dead specimens were found, and no living snails). Seen under a microscope, these tiny shells reveal their curious physique. They look like the corrugated pipe of a vacuum cleaner that’s been left tangled on the floor, with the end flared out like a tiny trumpet. These shells twist and turn, this way and that, as if they can’t decide which way to grow.

A few years later, Clements’ colleague Thor-Seng Liew finally tracked down live specimens of this tiny snail and set about studying them for his Ph.D, devising a theory to explain their bizarre coiling shapes. Liew suggested that the snails are doing their best to avoid getting eaten by predatory slugs. Retreating into their shells, the snails force their attackers to reach into an empty, bendy tube while their prospective dinner cowers at the end. The slugs’ proboscis simply can’t reach into such a deep and convoluted recess.

Meanwhile, Clements and other limestone enthusiasts have been campaigning to protect the remarkable but often overlooked places these snails come from. Being useless for agriculture or development, limestone hills were left more or less alone for a long time, but now cement companies are getting in on the act, razing them to the ground for the limestone inside. These are imperilled arks of biodiversity that few people have heard of. Year on year, hundreds of species are going extinct, most of them before they are discovered, when the hills they once lived on are taken away.

Compared to Clements and Liew’s bizarre find in the Malaysian hills, most shells are far less erratic in the way they grow, and indeed they are often quite predictable. For centuries, many great minds have contemplated the elegant sculptures and patterning of shells and wondered what might govern their construction. They have hunted for clues to explain the amazing realities and tempting possibilities of shells; they have probed ideas of what makes a shell work and which shapes may ultimately never show up; and they imagined that if they could find ways of drawing shells, if they could mimic what nature has been doing for eons, it would not only bring them closer to understanding how molluscs make their intricate homes, but they might also catch a glimpse of the origins of beauty itself. What many generations of mathematicians, artists, biologists and palaeontologists have found is unexpected and elegant: to construct an elaborate seashell – and decorate it – requires only a handful of rules.

Of all shell shapes, one of the simplest and most pleasing is the spiral of the chambered nautilus. The internal twist of these ocean-wanderers is revealed when their empty shells are sliced in two, from top to bottom. Trace the outer edge of a nautilus shell and you’ll see that it spins inwards in a very particular way. This graceful curve was among the first shapes in nature to be granted its own mathematical formula.

In the seventeenth century, French philosopher René Descartes composed a simple piece of mathematics for drawing a shape called the logarithmic spiral. Unlike an Archimedean spiral, which has whorls that are always spaced the same width apart, like a coiled snake, the gaps between successive whorls on a logarithmic spiral get increasingly wide. Logarithmic spirals flare open as they get bigger, just like a nautilus shell.

~~Spirals In Time : The Secret Life and Curious Afterlife of Seashells -by- Helen Scales

Day 136: Book Excerpt: The Chaos Imperative

Letters started pouring in to Switzerland’s University of Bern from physicists all over Europe with questions and praise. Some came from the most esteemed scientists of the day. The letters were addressed to one Albert Einstein, who a number of months earlier had published his theory of relativity. But what the letter writers didn’t know was that Einstein didn’t work at the university. The physicists knew that he lived in Bern and just assumed he was a professor at the university there.

In fact, Einstein had nothing to do with the university. He was a patent clerk. A government worker had turned the world of physics upside down.

We all know the story of how Einstein, at a young age, made stunning advances in physics. Most of us also have heard that Einstein was a poor student and was able to make his pioneering discoveries in physics despite being completely divorced from academia.

It was almost too extraordinary to believe. A twenty-six-year-old emerges seemingly out of nowhere with a scientific theory that changes the world. That alone would have been unprecedented. But ten years later Einstein once again revolutionized physics, reinventing our understanding of gravity. Today Einstein’s name is virtually synonymous with genius.

The explanation that most of us have grown up with for Einstein’s breakthroughs is that Einstein had such a brilliant and unusual mind that he—almost magically, in a stroke of insight—saw the universe in a whole new way.
In trying to understand Einstein’s unique genius, scientists over the years initially focused on the structural nature of his brain. Einstein had such an extraordinary mind, scientists reasoned, that there must be something fundamentally different about his brain.

When Einstein died in 1953, coroner Thomas Harvey removed what had become the most famous brain in history as a matter of course; it was a regular part of the autopsy procedure. What he did next, however—putting the brain in a jar of formaldehyde, slipping the jar into a bag, and walking off with it—was not. But Harvey believed it was his duty to science and to the world to preserve Einstein’s brain in order to let researchers study it and unlock the secrets of his mind.

In the succeeding years, neuroscientists, or neuroanatomists, as they used to be called, asked Harvey for certain sections of the brain in a race to pinpoint exactly which part of Einstein’s brain was so unique.

Scientists found that Einstein had a higher-than-average concentration of neurons in the part of the brain responsible for mathematical thinking. This seemed like a promising lead. The problem with this finding, however, was that Einstein wasn’t exceptionally gifted in math. His first wife, Mileva Maric, used to check all his calculations and correct them. And while Einstein was far more accomplished in math than your average English—or math—major, his discoveries weren’t really mathematical breakthroughs. Instead, his theories of relativity reconceptualized our notions of time and space. They were more a new set of ways of looking at the universe, supported by the math, than a set of complex mathematical formulas.

Another scientist, Marian C. Diamond, discovered that Einstein had more glial cells than average. Glial cells make up the myelin layer that insulates the brain’s axons, speeding up communication between the neurons. They also function as a distribution system, bringing energy to the neurons while removing waste.

However, only in one area of Einstein’s brain was the difference in glial cells statistically significant. And since Einstein’s brain was older than the other brains Diamond compared it against, and glial cells continue to divide as we age, it was only natural that Einstein had more of them. So while his glial network conceivably could have had something to do with his genius, we simply cannot know its impact for certain.
On and on went the physiological investigations. Scientists discovered that Einstein’s brain was wider than average. On the other hand, it also weighed less than average.

In the end, the studies on Einstein’s brain proved compromised in many ways and yielded no real insight into his genius. The reality is that each of us has unique idiosyncrasies in the makeup of our brain.

Even Einstein didn’t think it was his brain that made him who he was. He once commented that the gap between what the public thought of his intellectual prowess and the reality was “grotesque.”

But if it wasn’t his brain that made the difference, what did set Einstein apart? And what does Einstein’s genius have to do with chaos?

At the University of Zurich at the turn of the twentieth century, rows of well-dressed students would have been taking copious notes, smoking, and tackling complex formulas. One student who likely would not have been in the room, however, was Einstein, who was inclined to skip class and hang out in the coffeehouses on the Bahnhofstrasse, talking about new ideas in physics with the café crowd.

In the summer, while other physics students were working in labs or helping professors publish papers, Einstein hiked the beautiful trails of the Appenzell District in the Alps. It was as if his entire year were one big, unstructured interlude.

And that is our first clue to Einstein’s genius. To all appearances, Einstein was a slacker. Granted, he was a slacker obsessed with theoretical physics, but he was a slacker nonetheless.

He couldn’t be bothered to go to class. He engendered so little confidence in his academic abilities that one of his instructors suggested he give up studying physics altogether. In a great bit of irony, when graduation rolled around, Einstein was the only unemployed member of the class of 1900. His father, Hermann, tried to call in favors to get his son a job, but to no avail.

Imagine his poor mother’s desperate concern: “You have to start going to class.” “What happened to the intelligent young man I knew?” “You know, if you worked harder, you’d be surprised by how much progress you could make.”

It’s easy to sympathize with his parents’ likely responses. Einstein’s seemingly dilettante behavior would have driven most parents to distraction.

But his parents’ misgivings were for naught. What Einstein was actually doing was exercising a very special part of his brain.

Most of us tend to have clearly defined ideas about what makes up the road to success. We value discipline and diligence, hard work, and the idea of “paying your dues.” Unstructured time just “hanging out” is for teenagers with too much time on their hands, we think, and for surfer bums. Most of us need to pay attention, study hard, and learn.

But that’s not what Einstein did at all. As we’ll see, Einstein followed a specific process in developing his ideas—one intimately related to the chaos imperative. It is one that arguably led to his extraordinary and unpredictable brilliance. And it is one that each of us can tap into as well.

~~The Chaos Imperative: How Chance and Disruption Increase Innovation, Effectiveness and Success -by- Ori Brafman and Judah Pollack

Day 76 : Book Excerpt : Here’s Looking at Euclid

On the other hand, our approach to mathematics is very much influenced by culture. The selection of base ten, for example, was premised not on mathematical reasons but on physiological ones, the numbers of our fingers and toes. Language also shapes mathematical understanding in surprising ways. In the West, for example, we are held back by the words we have chosen to express numbers.

In almost all Western European languages, number words do not follow a regular pattern. In English we say twenty-one, twenty-two, twenty-three. But we don’t say tenty-one, tenty-two, tenty-three—we say eleven, twelve, thirteen. Eleven and twelve are unique constructions and even though thirteen is a combination of three and ten, the three part comes before the ten part—unlike twenty-three, in which the three part comes after the twenty part. Between ten and twenty, English is a mess.

In Chinese, Japanese and Korean, however, number words do follow a regular pattern. Eleven is written ten one. Twelve is ten two, and so on with ten three, ten four up to ten nine for 19. Twenty is two ten and 21 is two ten one. You pronounce numbers in all cases just as you see them written down. So what? Well, it does make a difference at a young age. Experiments have repeatedly shown that Asian children find it easier to learn to count than Europeans. In one study with Chinese and American four-and five-year-olds, the two nationalities performed similarly learning to count to 12, but the Chinese were about a year ahead with higher numbers. A regular system also makes arithmetic clearer to understand. A simple sum like 25 plus 32 when expressed as two ten five plus three ten two is one step closer to the answer already: five ten seven.
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We are also handicapped by how long it takes us to say numbers. In The Number Sense, Stanislas Dehaene writes down the list 4, 8, 5, 3, 9, 7, 6 and asks us to spend 20 seconds memorizing it. English speakers have a 50 percent chance of remembering the seven numbers correctly. By contrast, Mandarin Chinese speakers can memorize nine digits in this way. Dehaene says that this is because the number of digits we can hold in our heads at any one time is determined by how many we can say in a two-second loop. The Chinese words for one to nine are all concise single syllables: yi, er, san, si, wu, liu, qi, ba, jiu. They can be uttered in less than a quarter of a second each, so in a two-second span, a Chinese speaker can rattle through nine of them. English number words, by contrast, take just under a third of a second each to say (thanks to “seven,” with two syllables, and the extended syllable “three”), and so our limit in two seconds is seven. The record, however, goes to the Cantonese, whose digits are spoken with even more brevity. They can remember ten of them in a two-second period.

While Western languages seem to be working against any mathematical ease of understanding, in Japan, language is recruited as an ally. Words and phrases are modified in order to make their multiplication tables, called kuku, easier to learn. The tradition of these tables originated in ancient China, spreading to Japan around the eighth century. Ku in Japanese is nine, and the name comes from the fact that the tables used to begin at the end, with 9 × 9 = 81. Around 400 years ago they were changed so that the kuku now begins, “One one is one.”

The words of the kuku are simply:

One one is one
One two is two
One three is three . . .
This carries on to “One nine is nine,” and then the twos begin with:
Two one is two
Two two is four
And so on to nine nine is eighty-one.

So far, this seems very similar to the plain British style of reciting the times tables. In the kuku, however, whenever there are two ways to pronounce a word, the way that flows better is used. For example, the word for one can be in or ichi, and rather than starting the kuku with either in in or ichi ichi, the Japanese use the more sonorous combination in ichi. The word for eight is ha. Eight eights should be ha ha. Yet the line in the kuku for 8 × 8 is happa, since it rolls quicker off the tongue. The result is that the kuku is rather like a piece of poetry, or a nursery rhyme. When I visited an elementary school in Tokyo and watched a class of seven-and eight-year-olds practice their kuku, I was struck by how much it sounded like a rap—the phrases were syncopated and said with great animation. Certainly it bore no relation to how I remember reciting my times tables at school, which was with the metronomic delivery of a steam train going up a hill. Makiko Kondo, the teacher, said that she teaches her pupils kuku with an up-tempo rhythm because this makes it fun to learn. “First we get them to recite it, and only some time later do they come to understand the real meaning.” The poetry of the kuku seems to embed the times tables in Japanese brains. Japanese adults told me that they know, for example, that seven times seven is forty-nine not because they remember the math but because the music of “seven seven forty nine” sounds right.

~~Here’s Looking at Euclid -by- Alex Bellos

Day 40: Book Excerpt: The Information

Solomonoff, Kolmogorov, and Chaitin tackled three different problems and came up with the same answer. Solomonoff was interested in inductive inference: given a sequence of observations, how can one make the best predictions about what will come next? Kolmogorov was looking for a mathematical definition of randomness: what does it mean to say that one sequence is more random than another, when they have the same probability of emerging from a series of coin flips? And Chaitin was trying to find a deep path into Gödel incompleteness by way of Turing and Shannon—as he said later, “putting Shannon’s information theory and Turing’s computability theory into a cocktail shaker and shaking vigorously.” They all arrived at minimal program size. And they all ended up talking about complexity.

The following bitstream (or number) is not very complex, because it is rational:

D: 14285714285714285714285714285714285714285714285714…


It may be rephrased concisely as “PRINT 142857 AND REPEAT,” or even more concisely as “1/7.” If it is a message, the compression saves keystrokes. If it is an incoming stream of data, the observer may recognize a pattern, grow more and more confident, and settle on one-seventh as a theory for the data.

In contrast, this sequence contains a late surprise:

E: 10101010101010101010101010101010101010101010101013


The telegraph operator (or theorist, or compression algorithm) must pay attention to the whole message. Nonetheless, the extra information is minimal; the message can still be compressed, wherever pattern exists. We may say it contains a redundant part and an arbitrary part.

It was Shannon who first showed that anything nonrandom in a message allows compression:

F: 101101011110110110101110101110111101001110110100111101110


Heavy on ones, light on zeroes, this might be emitted by the flip of a biased coin. Huffman coding and other such algorithms exploit statistical regularities to compress the data. Photographs are compressible because of their subjects’ natural structure: light pixels and dark pixels come in clusters; statistically, nearby pixels are likely to be similar; distant pixels are not. Video is even more compressible, because the differences between one frame and the next are relatively slight, except when the subject is in fast and turbulent motion. Natural language is compressible because of redundancies and regularities of the kind Shannon analyzed. Only a wholly random sequence remains incompressible: nothing but one surprise after another.
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Even Π retains some mysteries:

C: 3.1415926535897932384626433832795028841971693993751…


The world’s computers have spent many cycles analyzing the first trillion or so known decimal digits of this cosmic message, and as far as anyone can tell, they appear normal. No statistical features have been discovered—no biases or correlations, local or remote. It is a quintessentially nonrandom number that seems to behave randomly. Given the nth digit, there is no shortcut for guessing the nth plus one. Once again, the next bit is always a surprise.

How much information, then, is represented by this string of digits? Is it information rich, like a random number? Or information poor, like an ordered sequence?

The telegraph operator could, of course, save many keystrokes—infinitely many, in the long run—by simply sending the message “Π.” But this is a cheat. It presumes knowledge previously shared by the sender and the receiver. The sender has to recognize this special sequence to begin with, and then the receiver has to know what Π is, and how to look up its decimal expansion, or else how to compute it. In effect, they need to share a code book.

This does not mean, however, that Π contains a lot of information. The essential message can be sent in fewer keystrokes. The telegraph operator has several strategies available. For example, he could say, “Take 4, subtract 4/3, add 4/5, subtract 4/7, and so on.” The telegraph operator sends an algorithm, that is. This infinite series of fractions converges slowly upon Π, so the recipient has a lot of work to do, but the message itself is economical: the total information content is the same no matter how many decimal digits are required.

The issue of shared knowledge at the far ends of the line brings complications. Sometimes people like to frame this sort of problem—the problem of information content in messages—in terms of communicating with an alien life-form in a faraway galaxy. What could we tell them? What would we want to say? The laws of mathematics being universal, we tend to think that Π would be one message any intelligent race would recognize. Only, they could hardly be expected to know the Greek letter. Nor would they be likely to recognize the decimal digits “3.1415926535 …” unless they happened to have ten fingers.

~~The Information -by- James Gleick