Ladybird week

This is a piece which will find its way into a book that is just about ready to offer to publishers, under the working title Looking at Small Things.

The book has its roots in a time, 68 years ago, when my profoundly unscientific parents bought me a toy microscope, made of Bakelite, with plastic lenses. I was entranced, but I could find very little to look at, though I did find a ferocious-looking beast on a Dahlia.

Somehow, I managed to learn that this was the larva of a ladybird, but as I could learn no more, and found no more exciting animals in the garden, so I moved onto other things, but a couple of weeks back, they were in my garden again, and I showed them to my twin granddaughters, but it took me a few days to recall what they were.

Over the years, a number of my books have been about finding interesting things to look at, how to catch them, how to keep them safely (for both the keeper and the keepee). I do this, because I had zero support or encouragement, and that simply wasn't right. To give my father some credit, he once showed me an adult cicada emerging from the pupa case, but there was precious little engagement of showing how involved.

My books, like Exploring the Environment and Australian Backyard Naturalist (both out of print, but try libraries) are designed to give adult guides and independent readers the ways and means to do that.

Now, close to ten times my age then, I had an approach from a start-up at Flinders University who had deduced  that I was interested in stuff like that. The upshot was that I told them I would write a guide for them as a pro bono, keeping the rights to any ensuing book. They make a very neat little gadget that clips onto phones and tablets. It is better used on really tiny things, but the second shot above was taken with it.

You can download a free version of a draft of my new book for free, using this link, but it is slanted towards teachers, and is targeted at helping them make sense of the kludgey and totally inadequate "National Science Curriculum".  When the greatly revised  Looking at Small Things comes out, it will be far more child-friendly.


Anyhow, back to my story: the ladybird larvae were everywhere. But then, I notice something: they were starting to pupate on the walls of our courtyard.

At first, I was a bit slow to catch on, but they kept coming in waves, so I got the whole life cycle thing. Here, you can see a number of steps along the way.

In the end, they emerged as adults. Knowing what to look for, maybe you can find these (or similar) beasties for yourself. To get a handle on the scale, the larvae were about 8 mm long, and the beetles below were about 4 mm from side to side.

The Speewah Girls’ snake circus

There weren’t that many women on the Speewah, but the ones that were there made up for it by the great ideas they used to have. Take the time Smiling Annie’s daughter Alice and Greasy Smith’s second youngest, Gertie decided to take some Speewah snakes down to the big smoke and put on a circus.

They’d thought about doing some acts themselves, but they decided their usual party tricks like riding a bicycle with three rolls of barbed wire and six loose melons was too ordinary. They tried to get Mick to do a strongman act for them, but he reckoned they’d be better off with snakes, because city folk are both scared of and fascinated by snakes.

The first thing Alice and Gertie did was to sit down and plan the acts they could use. First up they had some adding adders, where you would ask an easy sum, and the snakes would stick up enough heads over the side of the container to give you the answer. It was a fake, of course, because adders are deaf, and couldn’t hear the question, but Alice had a pup, the runt of one of the litters sired by Mick’s dog, and it could hear all right.

So it’d listen to the question, then nose enough of the adders, which would stick their heads up, rather than get nipped on the tail by the pup if they didn’t do it right. But even if it was a fake, the customers wouldn’t know it was just a dog doing the sums, and so they’d be impressed.

The next thing they decided on was a snaky equivalent of a lion-taming act, and for this, they decided to use a young python they found eating scrub bulls in the back paddock. What happened was they were looking for a horse that had gone missing, and they thought this python might know something about it, so they reckon Alice ripped its jaws open, and Gertie stepped inside, but all there she could see was a scrub bull that was bellowing and roaring for all it was worth.

In truth, Smiling Annie was there as well, and she held the snake’s tail, and the next bit was her idea. “Let’s see,” she says. “The tent’s only got four ‘roo hides in it, so you can fit about four hundred people, but you’ll never get all of that snake into the ring. It’d be best if you train it to open its jaws, then you can bring the front end in, just after you’ve fed it a bull, because snakes don’t roar, but your customers won’t know that the roar they hear is coming from the bull, not the snake.”

Gertie being the small one, she got the job of being the tamer of what they now called “Grendel, the world’s biggest worm”, which was a bit of a fake, seeing as how it was really a python, and people sort of knew that worms didn’t have two-metre teeth, but it still looked real impressive. Mind you, they could see a problem if they had to do matinees, because it took Grendel a full day to digest a bull, but in the end that wasn’t a problem.

Next up, they decided, was a high-wire act. That was easy, because they got some of the plaiting snakes. These are the only little snakes that can stand up to the big snakes in the back paddock, and that’s because they plait themselves together into a whip, and lash any big snake that comes near them, and they’re highly intelligent.

Anyhow, Annie rounded some up and explained what was on offer: a chance to see a bit of the country, free milk, plenty of frogs, and a chance to give Grendel a free lashing at any matinee performance. Of course, they’d need fancy uniforms, but the rest of the plaiting snakes had a whip-around, and in no time at all, they had lashings of cash.

The thing is, Grendel wasn’t too happy about the idea, as he’d had a few encounters with plaiting snakes, even in his young life, but that was no problem. The girls just got Smiling Annie to come around and smile at Grendel, and he decided that the whole idea had a lot of merit, and it was only for matinees.

Anyhow, the plaiting snakes were ideal for the high wire, but they worked themselves into a bigger routine, where they started out as a trapeze act, and swung back and forth, adding more snakes to the plait, then whipping up to tie off on the other post. It was a mistake for the girls to agree to this, because the plaiting snakes used this as an excuse to get more of their family into the show, and that was the first element of the disaster that was to come.

There was another problem when they tried to get some drop bears to ride tiptail snakes. These tiptails are completely harmless snakes, which only eat wild grapes and spinifex seeds, but when the wind gets up, the seeds blow around pretty fast, so the tiptails need to be even faster, and they rear up and race along on just the tip of their tail, cutting down on friction.

Well the drop bears would ride the tiptail snakes all right, but the first time the snakes reared up on their tails, two of the bears went feral, and bit the snakes on the neck. And even though Mrs Greasy Smith had filed down the bears’ teeth for them, it still hurt the tiptails.

Now I know I said the tiptails are harmless, but they also have a very mean streak and a nasty sense of humour, especially when something annoys them. I’ve seen more than one horse rider chased by tiptails after taking a horse over a tiptail nursery, and there’s nothing more upsetting than galloping full speed, and having four hissing snakes either side of you, four more behind you, and a couple of small ones jumping over you from side to side.

But while you can bluff a horse rider, drop bears have no imagination at all, so what the tiptails did was to race around the practice ring, faster and faster, and then lean out and bash the drop bears against the poles. So given the time it took to catch a drop bear alive and file its teeth, it just wasn’t worth it, so the tiptails were reduced to doing gymnastics and precision high diving, but people had seen all that before.

Flash Jack reckoned they ought to get the tiptails riding the drop bears, saying they could call it bear-back riding, but the girls wouldn’t be in it. Anyhow, Flash Jack had been telling the girls about hoop snakes for years, and they were never sure whether he was having a lend of them or not. So now they put the hard word on him to deliver some, and he had to admit that there weren’t any such animals.

That was no problem to Gertie. She had gone out and collected four young taipans — had to kill the mother, of course, but she got the young ones before they knew they were snakes, and brought them up with another litter of pups, fairly bright little pups they were, too, second cousins of Mick’s dog, and the snakes grew up thinking they were dogs.

But as cattle workers, the taipans were a dead loss, because every time they nipped a bull in the heels, it’d die. No worries, though, Gertie took them and trained them to hold their tails carefully in their mouths, with the poison fangs either side of the tail. Then she helped them get upright, and tried to get them to hoop along, but they just couldn’t manage it, so all the girls could do in the end was run them down a ramp and across the ring, or wheel them around the ring.

The juggling snakes were pretty good as well, and the strong snake act was Grendel’s tail, coming in through a flap in the roof — brought the house down once or twice until they got the cross-bracing right, and the snakes on unicycles were brilliant. The snake charming wasn’t much good though, as they had some of the adding snakes playing a tuba between them, two on the mouthpiece, and one on each key with Gertie coming out of the basket, but they forgot that all the adders were deaf, so nobody enjoyed it much, except the snakes.

But in the end, the whole show went broke. You see, you can’t really have a circus without clowns, and there’s just no way you can keep a red nose on a snake, because the elastic kept slipping off. So after all that effort, Alice and Gertie had to let the snakes go back into the bush again, where all of the snakes, including the adders, multiplied.

Still, circus training dies hard, and even today, you can find cooperative groups of plaiting snakes driving scrub bulls into the mouths of a large old python in the Speewah back paddock, assisted by a couple of taipans which sometimes seem to let out just the hint of a yelp. You’ll know the python straight off, as he’s only got one tooth left. And you’ll find these adders that pop their heads up over a log to look at you if you shout out a sum, but you have to shout real loud. So I suppose the snakes got something out of it, even if the girls didn’t.

Mick was able to use the tent, though. He turned it over, put loops around the base, and used it as a dilly-bag to carry his spare shears and a bit of a snack when he was heading off somewhere, and Gertie and Alice took Greasy’s second bullock team out on the road for a spell till they got over their disappointment. It was hard on the bullocks though, because Greasy just said to take them out on the road, and they assumed he meant them to carry the bullocks and the bullocks got embarrassed.

But that’s another story.

Can we trust statistics?

This chapter began as two radio talks delivered on the ABC, almost thirty years ago. My friend Peter Chubb asked me if I had addressed these issues, and I said that I hadn’t, but that I had provided a link to the text of the talk. Two nights later, I decided to add it, the next night, I rewrote it.

There is enough information here to let readers try the following exercise in Evil Statistics out for themselves.

*

Boris, Don and Tony went fishing, and caught ten fish. Four weighed 1 kg, two were 2 kg, two were three kg, one was 6 kg, and one was 10 kg. They reported that the average was 1 kg, 2 kg and 3 kg, and all were telling a sort of truth. Boris reported the mode, the most common mass, Don reported the median, the mass of the middle two fish, Tony reported the mean, adding all the masses and dividing by ten. Each value was true, each was different.

It all sounds a bit like “Lies, damned lies, and statistics”, but who first said that? The popular myth is that it was Mr. Disraeli, the well-known politician, but many quite reputable and reliable reference books blame author Mark Twain.

It turns out that it was first published by Twain all right, but Twain attributed the line to Disraeli, and you won't find the story in any earlier publication than Twain's autobiography. In short, Mark Twain made the whole thing up! Disraeli never spoke those words: Twain invented them all, but he wanted the joke to have a greater force, and so gave the credit to an English politician.

Twain wasn't only well-known for his admiration of a good “Stretcher” (of the truth, that is), he even lied when he was talking about lies, and his name wasn't even Mark Twain, but Samuel Clemens! Now would you buy a used statistic from this man?

Last century, when Disraeli is supposed to have made the remark, statistics were just numbers about the State. The state of the State, all summed up in a few simple numbers, you might say.

Now governments being what they are, or were, there was more than a slight tendency in the nineteenth century to twist things just a little, to bend the figures a bit, to bump up the birth rate, or smooth out the death rate, to fudge here, to massage there, to adjust for the number you first thought of, to add a small conjecture or maybe to slip in the odd hypothetical inference.

It was all too easy to tell a few small extravagances about one's armaments capacity, or to spread the occasional minor numerical inexactitude about whatever it was rival nations wanted to know about, and people did just that. Even today, when somebody speaks of average income, if you don’t smell fish, at least remember them, and ask if that’s the mean, the median or the mode.

When I was young, I smoked cigarettes, but the cost and the health risks convinced me, so I stopped, back in 1971. Smokers think we reformed smokers are tiresome people who keep on at them, trying to get them to stop as well.

The non-smokers say those who still puff smoke are the tiresome people, who can't see the carcinoma for the smoke clouds, who deny any possibility of any link between smoking and anything. Like the tobacco pushers, the smokers dismiss the figures contemptuously as “only statistics”. The really tiresome smoker will even say a few unkind things about the statisticians who are behind the figures. Or about the statisticians who lie behind the figures.

By the end of the 19th century, statistics were no longer the mere playthings of statesmen, they were way to clump large groups of related facts into convenient chunks. If you can see how the statistics were arrived at, perhaps you can trust them.

At one stage in my career, I led a gang of people who gathered statistics and messed about with numbers, but we preferred to be called ‘number-crunchers’. People say a statistician is “somebody who's rather good around figures, but who lacks the personality to be an accountant”.

They speak of the statistician who drowned in a lake with an average depth of 15 cm. We are told that a statistician collects data and draws confusions, or draws mathematically precise lines from an unwarranted assumption to a foregone conclusion. They say “X uses statistics much as a drunkard uses a lamp-post: rather more for support than for illumination”.

Crusty old conservatives give us a bad name, pointing out that tests reveal that half our nation's school leavers to be below average, which is true, but it is equally true that the vast majority of Australians have more than the average number of legs. All you need is one Australian amputee!

If somebody does a Little Jack Horner with a pie that's absolutely bristling with statistical items and they produce just one statistical plum, I won't be impressed at all: the plum's rather more likely to be a lemon, anyhow. 

The statistics have to be plausible and significant. Later, I will show you a statistical link between podiatrists and public telephones: this is obviously nonsense, and we will ignore it. There is no logical reason for either to influence the other.

Still, unless there is a plausible reason why X might cause Y, it's all very interesting, and I'll keep a look-out, just in case a plausible reason pops up later, but I won't rush to any conclusion. Not just yet, I won't.

First, I will check on the likelihood of a chance link, something we call statistical significance. After all, if somebody claims to be able to tell butter from margarine, you wouldn't be too convinced by a single successful demonstration, would you? Well, perhaps you might be convinced: certain advertising agencies think so, anyway.

If you tossed a coin five times, you wouldn't think it meant much if you got three heads and two tails, unless you were using a double-headed coin, maybe. If somebody guessed right three, or even four, times out of five, on a fifty-fifty bet, you might still want more proof.

You should, you know, for there's a fair probability it was still just a fluke, a higher probability than most people think. There's about one chance in six of correctly guessing four out of five fifty-fifty events. Here is a table showing the probabilities of getting zero to five correct from five tosses:

zero right	one right	two right	three right	four right	five right
1/32	5/32	10/32	10/32	5/32	1/32

 The clever reader may notice a resemblance to Pascal’s triangle here!

Now back to the butter/margarine study. Getting one right out of one is a fifty-fifty chance, while getting two right out of two is a twenty five per cent chance, still a bit too easy, maybe. So you ought to say “No, that's still not enough. I want to see you do it again!”.

Statistical tests work in much the same way. They keep on asking for more proof until there's less than one chance in twenty of any result being just a chance fluctuation. The thing to remember is this: if you toss a coin often enough, sooner or later you'll get a run of five of a kind.

As a group, scientists have agreed to be impressed by anything rarer than a one in twenty chance, quite impressed by something better than one in a hundred, and generally they're over the moon about anything which gets up to the one in a thousand level. That's really strong medicine when you get something that significant.

There. Did you spot the wool being pulled down over your eyes, did you notice how the speed of the word deceives the eye, the ear, the brain and various other senses? Did you feel the deceptive stiletto, slipping between your ribs? We test statistics to see how “significant” they are, and now, hey presto, I'm asserting that they really are significant. A bit of semantic jiggerypokery, in fact.

And that's almost as bad as the sort of skullduggery people get up to when they're bad-mouthing statistics. Even though something may be statistically significant, that's a long way away from the thing really being scientifically significant, or significant as a cause, or significant as anything else, for that matter.

Statistics make good servants but bad masters. We need to keep them in their places, but we oughtn't to refuse to use statistics, for they can serve us well. Now you are ready to object when I assert that all the podiatrists in New South Wales seem to be turning into public telephone boxes in South Australia, and it all began with Florence Nightingale. Most people think of her as the founder of modern nursing, but as part of that she created ways to use statistics to pinpoint facts.

After her name was made famous, directing nursing in the Crimean war, she returned to London in 1857, and started to look at statistics, and the way they were used. She wrote a pamphlet called “Mortality in the British Army”, and the very next year, she was elected to the newly formed Statistical Society.

She looked at deaths in hospitals, and demanded that they keep their figures in the same way. The Statistical Congress of 1860 had, as its principal topic, her scheme for uniform hospital statistics. It isn’t enough to say Hospital X loses more patients than Hospital Y does, so therefore Hospital X is doing the wrong thing.

We need to look at the patients at the two hospitals, and make allowances for other possible causes. We have to study the things, the variables, which change together. Statistics, remember, are convenient ways of wrapping a large amount of information up into a small volume. A sort of short-hand condensation of an unwieldy mess of bits and pieces.

And one of the handiest of these short-hand describers is the correlation coefficient, a measure of how two variables change at the same time, the one with the other. Now here I'll have to get technical for a moment. You can calculate a correlation coefficient for any two variables, things like number of cigarettes smoked, and probability of getting cancer.

The correlation coefficient is a simple number which can suggest how closely related two sets of measurements really are. It works like this: if the variables match perfectly, rising and falling in perfect step, the correlation coefficient comes in with a value of one. But if there's a perfect mismatch, where the more you smoke, the smaller your chance of surviving, then you get a value of minus one.

With no match at all, no relationship, you get a value somewhere around zero. But consider this: if you have a whole lot of golf balls bouncing around together on a concrete floor, quite randomly, some of them will move together, just by chance.

There’s no cause, nothing in it at all, just a chance matching up. And random variables can match up in the same way, just by chance. And sometimes, that matching-up may have no meaning at all. This is why we have tests of significance. We calculate the probability of getting a given correlation by chance, and we only accept the fairly improbable values, the ones that are unlikely to be caused by mere chance.

We aren’t on safe ground yet, because all sorts of wildly improbable things do happen by chance. Winning the lottery is improbable, though the lotteries people won't like me saying that. But though it's highly improbable, it happens every day, to somebody. With enough tries, even the most improbable things happen.

So here's why you should look around for some plausible link between the variables, some reason why one of the variables might cause the other. But even then, the lack of a link proves very little either way. There may be an independent linking variable.

Suppose smoking was a habit which most beer drinkers had, suppose most beer drinkers ate beer nuts, and just suppose that some beer nuts were infected with a fungus which produces aflatoxins that cause slow cancers which can, some years later, cause secondary lung cancers.

In this case, we'd get a correlation between smoking and lung cancer which still didn't mean smoking actually caused lung cancer. And that's the sort of grim hope which keeps those drug pushers, the tobacco czars going, anyhow. It also keeps the smokers puffing away at their cancer sticks.

It shouldn't, of course, for people have thrown huge stacks of variables into computers before this. The only answer which keeps coming out is a direct and incontrovertible link between smoking and cancer. The logic is there, when you consider the cigarette smoke, and how the amount of smoking correlates with the incidence of cancer. It's an open and shut case.

I'm convinced, and I hope you are too. Still, just to tantalise the smokers, I'd like to tell you about some of the improbable things I got out of the computer in the 1980s. These aren't really what you might call damned lies, and they are only marginally describable as statistics, but they show you what can happen if you let the computer out for a run without a tight lead.

Now anybody who's been around statistics for any time at all knows the folk-lore of the trade, the old faithful standbys, like the price of rum in Havana being highly correlated with the salaries of Presbyterian ministers in Massachusetts, and the Dutch (or sometimes it's Danish) family size which correlates very well with the number of storks' nests on the roof.

More kids in the house, more storks on the roof. Funny, isn't it? Not really. We just haven't sorted through all of the factors yet. The Presbyterian rum example is the result of correlating two variables which have increased with inflation over many years.

You could probably do the same with the cost of meat and the average salary of a vegetarian, but that wouldn't prove anything much either. In the case of the storks on the roof, large families have larger houses, and larger houses in cold climates usually have more chimneys, and chimneys are what storks nest on. So naturally enough, larger families have more storks on the roof. With this information, the observed effect is easy to explain, isn't it?

There are others, though, where the explanation is less easy. Did you know, for example, that Hungarian coal gas production correlates very highly with Albanian phosphate usage? Or that South African paperboard production matches the value of Chilean exports, almost exactly?

Or did you know the number of iron ingots shipped annually from Pennsylvania to California between 1900 and 1970 correlates almost perfectly with the number of registered prostitutes in Buenos Aires in the same period? No, I thought you mightn't.

These examples are probably just a few more cases of two items with similar natural growth, linked in some way to the world economy, or else they must be simple coincidences. There are some cases, though, where, no matter how you try to explain it, there doesn't seem to be any conceivable causal link. Not a direct one, anyhow.

There might be indirect causes linking two things, like my hypothetical beer nuts. These cases are worth exploring, if only as sources of ideas for further investigation, or as cures for insomnia. It beats the hell out of calculating the cube root of 17 to three decimal places in the wee small hours, my own favourite go-to-sleep trick.

Now let's see if I can frighten you off listening to the radio, that insomniac's stand-by. Many years ago, in a now-forgotten source, I read that there was a very high correlation between the number of wireless receiver licences in Britain, and the number of admissions to British mental institutions.

At the time, I noted this with a wan smile, and turned to the next taxing calculation exercise, for in those far-off days, all correlation coefficients had to be laboriously hand-calculated. It really was a long time ago when I read about this effect.

It struck me, just recently, that radio stations pump a lot of energy into the atmosphere. In America, the average five-year-old lives in a house which, over the child's life to the age of five, has received enough radio energy to lift the family car a kilometre into the air. That's a lot of energy.

Suppose, just suppose, that all this radiation caused some kind of brain damage in some people. Not all of them necessarily, just a susceptible few. Then, as you get more licences for wireless receivers in Britain, so the BBC builds more transmitters and more powerful transmitters, and more people will be affected. And so it is my sad duty to ask you all: are the electronic media really out to rot your brains? Will cable TV save us all?

Presented in this form, it's a contrived and, I hope, unconvincing argument. Aside from anything else, the radiation is the wrong wave-length and cannot change any cells. My purpose in citing these examples is to show you how statistics can be misused to spread alarm and despondency. But why bother?

Well, just a few years ago, problems like this were rare. As I mentioned, calculating just one correlation coefficient was hard yakka in the bad old days. Calculating the several hundred correlation coefficients you would need to get one really improbable lulu was virtually impossible, so fear and alarm seldom arose.

That was before the day of the personal computer and the hand calculator. Now you can churn out the correlation coefficients faster than you can cram the figures in, with absolutely no cerebral process being involved.

As never before, we need to be warned to approach statistics with, not a grain, but a shovelful, of salt. The statistic which can be generated without cerebration is likely also to be considered without cerebration. Which brings me, slowly but inexorably to the strange matter of the podiatrists, the public telephones, and the births.

Seated one night at the keyboard, I was weary and ill at ease. I had lost one of those essential connectors which link the parts of one's computer. Then I found the lost cord, connected up my computer, and fed it a huge dose of random data.

I found twenty ridiculously and obviously unrelated things, so there were one hundred and ninety correlation coefficients to sift through. That seemed about right for what I was trying to do.

When I was done, I switched on the printer, and sat back to wait for the computer to churn out the results of its labours. The first few lines of print-out gave me no comfort, then I got a good one, then nothing again, then a real beauty, and so it went: here are my cunningly selected results. I have simply used, for good reasons, the methods of the crooks and con-men.

	Tasmanian birth rate	SA public phones	NSW podiatrist registrations
Tasmanian birth rate	1	+0.94	-0.96
SA public phones	+0.94	1	-0.98
NSW podiatrist registrations	-0.96	-0.98	1

Well of course the podiatrists and phones part is easy. Quite clearly, New South Wales podiatrists are moving to South Australia and metamorphosing into public phone boxes. Or maybe they're going to Tasmania to have their babies, or maybe Tasmanians can only fall pregnant in South Australian public phone booths.

Or maybe codswallop grows in computers which are treated unkindly. Figures can't lie, but liars can figure. I would trust statistics any day, so long as I can find out where they came from, and I'd even trust statisticians, so long as I knew they knew their own limitations. Most of the professional ones do know their limitations: it's the amateurs who are dangerous.

I'd even use statistics to choose the safest hospital to go to, if I had to go. But I'd still rather not go to hospital in the first place. After all, statistics show clearly that more people die in the average hospital than in the average home.

The original version is here:

http://members.ozemail.com.au/~macinnis/ockhams/stats.htm.

Science Playwiths 1.

I am about to sign a contract for a new book which I want to be called Science Playwiths, because it originated in a long-running web site of that name. It covers STEAM: Science, Technology, Engineering, Arts and Mathematics. Here's a first sample, about curious number facts.

 *

One estimate of the size of the entire universe puts its radius at 3 x 10²³ times larger than the size of the observable universe. That is almost exactly half the value of Avogadro’s number, which every good chemist knows to be 6.022 x 10²³. So what?

The speed of light in terafurlongs per fortnight is 1.803, close enough for government work to the metric equivalent of a fathom, showing that any measured value can be given almost any number by a cunning choice of units. Any reasoning based on the coincidence of two values needs to be questioned closely to see if the coincidence is just, well, a coincidence—or the work of somebody using peculiar units to get a result.

For example, the number of islands in the Hawaiian island chain is 137, and the ratio 1/137, often referred to as alpha, is the fine structure constant in physics. This value represents the probability that an electron will emit or absorb a photon. It is the square of the charge of the electron divided by the speed of light times Planck’s constant, and it is just a number: there are no dimensions or units involved at all.

The significance of alpha was first spelled out in 1915 by a physicist named Arnold Sommerfeld—at the time, measurement errors made the value closer to 136—and physics ever since has been littered with efforts to explain the number.

The most famous attempt was that of Sir Arthur Eddington, a prominent astronomer who believed that such constants could be used to produce a theory of the universe. He built a huge 16-dimensional equation full of these constants and claimed that alpha could be calculated from the number of terms: (16² - 16)/2 + 16, or 136.

Unfortunately, experiments quickly showed that alpha was really closer to 137. Eddington was not dismayed. He said he had forgotten to add one more factor, alpha itself, and made the value 137. For this, Punch magazine dubbed him Sir Arthur Adding-One.

Eddington was not deterred. Proudly he proclaimed that the firmament contains exactly (137 - 1) x 2²⁵⁶ protons. In 1938, he declared:

I believe there are 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 protons in the universe and the same number of electrons.

Of course, he may have been right; I have not yet been able to count them all, and it’s hard enough trying to find the value of 2²⁵⁶.