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Wednesday, 27 May 2015

Adventures with 1729


Talking of cubes as I was in my last post, even when people are thinking about the same thing, they are probably thinking in different ways. A famous English mathematician called Hardy once visited a very clever Indian mathematician called Ramanujan in hospital.

He probably wanted to know if Ramanujan had discovered a curious set of numbers called Carmichael numbers, so Hardy said casually that he had travelled there in a taxi with a boring number, 1729.

Ramanujan sat up in bed. "Oh no, Hardy, on the contrary, that is a very interesting number. It is the first number which is the sum of two cubes in two different ways!"

When Hardy thought it through, he realised that 1729 was the sum of 9 cubed and 10 cubed (729 + 1000), or of 12 cubed and 1 cubed (1728 + 1). When I came across this story, I noticed something much less interesting: 1729 is the product of three prime numbers which are in mathematical progression (7 x 13 x 19) -- I guess I am just a bit boring where numbers go.

One day, I had an idea for a problem: using just the numbers 1, 7, 2 and 9, in that order, and using any mathematical notation, I set out to see how many different numbers I could get.  I began with trial and error, a good old-fashioned method that has always served humanity well.  Bang the rocks together, notice what makes useful flakes, and bang them again, that sort of thing.

That got me a few answers, like 1 = 1^729, or 1 = 1 + 7 + 2 - 9 or 6 = 1 + (7x2) -9 but then I began running out of new numbers.  Then I switched to the sort of clever trick that is used when pharmacists take drugs that work, and fiddle with them, changing the bits systematically.

You can find some hints by poking around in this area. DO look around, or you will miss some important stuff.
 
Then we could turn to the problem that I will describe in my next post, and look at the tabulation method that I use there. . .

Friday, 22 May 2015

Calculating cube roots in your head

I thought I would swing over to mathematics for a bit.

You don't need a real aptitude to enjoy maths, and even if you had woefully useless teachers (as I did),  you can still have fun with numbers.


Are you one of those people who play with numbers? I was, even was I was small, and when I was supposed to be sleeping, I would be lying in bed, going "2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 . . ." on and on, seeing how high I could go.

I usually got 22 or 23 terms before I lost track and had to start again. I just enjoy looking for patterns (or I fell asleep).

Sometimes, I find something useful, like a way to fool people into thinking that I had memorised the cubes of all the numbers from 100 to 200.

I developed this trick by looking at the final digit of various cubes:

If you cube a number ending in 1 the result ends in 1
If you cube a number ending in 2 the result ends in 8
If you cube a number ending in 3 the result ends in 7
If you cube a number ending in 4 the result ends in 4
If you cube a number ending in 5 the result ends in 5
If you cube a number ending in 6 the result ends in 6
If you cube a number ending in 7 the result ends in 3
If you cube a number ending in 8 the result ends in 2
If you cube a number ending in 9 the result ends in 9
and obviously,
If you cube a number ending in 0 the result ends in 0

See the pattern? If you know the last digit of the cube, you know the last digit of the seed number that was cubed.

Then I realised that you can memorise the approximate ranges for the cubes of 100 to 109, 110 to 119 and so on:
100 – 109 1 to 1.3 million
110 – 119 1.3 to 1.7 million
120 – 129 1.7 to 2.2 million
130 – 139 2.2 to 2.7 million
140 – 149 2.7 to 3.3 million
150 – 159 3.4 to 4 million
160 – 169 4.1 to 4.8 million
170 – 179 4.9 to 5.7 million
180 – 189 5.8 to 6.7 million
190 – 199 6.8 to 8 million


so if I hear 1442897, I know from the first two digits that we are between 110 and 119, and the last digit of the cube (7) tells me the last digit of the number being cubed is 3, so the answer is 113.

Try it!

Saturday, 9 May 2015

Curtiosity about atoms



First, a note about "curtiosity": it is the key attribute of Rudyard Kipling's 'Elephant's Child. It isn't a typo. The curtiosities were collected originally as possible epigraphs, and then as a possible book in their own right, but it all seemed too hard.

Most collections of "quotes" on the web lack the necessary details of chapter and verse and are commonly spurious. You get my notes, so you know where they came from, and any agile mind will recognise the occasional traps laid for mindless plagiarists. Verb. sap. (Look that up, if you need to!)

Mind you, the lack of a full source may just mean I was busy the day I collected that quote, but such information as I give should be reliable.

Atoms move in the void and catching each other up jostle together, and some recoil in any direction that may chance, and others become entangled with one another in various degrees according to their shapes and sizes and positions and orders, and they come together and thus the coming into being of composite things is effected.
— Simplicius (c. 400 BC), De Caelo

To understand the very large, we must understand the very small.
— Democritus (470 - 380 BC

The Atoms of Democritus
And Newton's Particles of Light
Are sands upon the Red sea shore
Where Israel's tents do shine so bright.
— William Blake (1757 - 1827), Complete Blake (Oxford Paperback, 1974), page 418.

1. From nothing comes nothing. Nothing that exists can be destroyed. All changes are due to the combination and separation of molecules.
2. Nothing happens by chance. Every occurrence has its cause from which it follows by necessity.
3. The only existing things are atoms and empty space; all else is mere opinion.
4. The atoms are infinite in number and infinitely various in form; they strike together and the lateral motions and whirlings which thus arise are the beginnings of worlds.
5. The varieties of all things depend upon the varieties of their atoms, in number, size, and aggregation.
6. The soul consists of fine, smooth, round atoms like those of fire. These are the most mobile of all. They interpenetrate the whole body and in their motions the phenomena of life arise.
— Robert Andrews Millikan quotes these (translated) words of Democritus in his book The Electron, saying that they are from [Sir John] Tyndall.

When any body exists in the elastic state, its ultimate particles are separated from each other to a greater distance than in any other state; each particle occupies the centre of a comparatively large sphere, and supports its density by keeping all the rest, which by their gravity or otherwise, are disposed to encroach upon it, at a respectable distance.
Chemical analysis and synthesis go no further than to the separation of particles one from another, and to their reunion. No new creation or destruction of matter is within the reach of the chemical agency. We might as well attempt to introduce a new planet into the solar system, or to annihilate one already in existence, as to create or destroy a particle of hydrogen. All the changes we can produce consist in separating particles that are in a state of cohesion or combination, and joining those that were previously at a distance.
— John Dalton, A New System of Chemical Philosophy, 1808.

They may say what they like. Everything is organised matter.
— Napoleon Bonaparte (1769 - 1821

We shall never get people whose time is money to take much interest in atoms.
— Samuel Butler (1835 - 1902), Notebooks

The first support of the isotope theory among non-radioactive elements was given by the anomalous behaviour of the inactive gas neon, when analysed by Sir J. J. Thomson's method of positive rays . . . This peculiarity was that whereas all elements previously examined gave single, or apparently single, parabolas, that given by neon was definitely double. The brighter curve corresponded roughly to an atomic weight of 20, the fainter companion to one of 22, the atomic weight of neon being 20.20.
— Francis Aston (1877 - 1945), address before the Royal Institution, 1921. This established the existence of isotopes.

No one has ever seen, nor probably ever will see, an atom, but that does not deter the physicist from trying to draw a plan of it, with the aid of such clues to its structure as he has.
— Maria Goeppert Mayer (1906 - 1972), 'The Structure of the Nucleus', Scientific American Reader (1953), page 116.

In fact it may be logically impossible for anyone to be able to correctly visualize certain physical systems, such as atoms, because they contain features that simply do not exist in the world of our experience.
— Paul Davies, The Mind of God, Penguin Books, 1990, p. 18.

There have been almost innumerable attempts to reduce the differences between atomic weights to regularity by contriving some formula which will express the numbers which represent the weights with all their irregularities. Needless to say, such attempts have in no way been successful.
— Sir William Ramsay (1852 - 1916), address to the British Association, Toronto, 1897.

Wednesday, 6 May 2015

A question of collaboration



Anybody who knows much about the books I write could be excused for sniffing and saying "Ah, yes, Macinnis—always got his nose stuck in the 19th century!"  Well. it's true—I do find the middle of the 19th century fascinating, but only as a nice place to visit.  I wouldn't want to live there, not unless I can have a time machine to send back antibiotics, good cheese and a decent bicycle.

It was a simple era, and somebody from our time can understand even the fastest of the changes in science and technology happening back then.  Later, it all became too complex.  Mark Twain's Connecticut Yankee could go back in time to King Arthur's court and reproduce 19th century technology, but could you make a transistor, a computer, or a motor car?

This belongs to somebody, If it's yours, give me a shout!
I don't long for the old times, but I find them interesting, and I like finding things that belong more in the realm of alternative-future fiction or steam punk.  I lap up clockwork fly traps, steam-powered, wing-flapping flying machines, balloons powered by tanks of compressed hydrogen driving engines that turn the propellers or even balloons hauled by eagles.

It's hard not to feel smugly superior, reading their predictions of the future.  Let me lie back on the couch, doctor, and I'll tell you about it.

It all began when I was a teenager, still in short pants, reading the words of Egon Larsen, a mid-20th century science writer, praising the virtues of atomic energy:

"Within a few years isotopes will turn up in many more expected or unexpected places — perhaps the slogan 'Gamma Washes Whiter', will become quite familiar to us when our ultra-sonic washing machines are equipped with some gamma source to sterilize shirts and socks and napkins." [1]
A little earlier, he'd explained nuclear waste disposal like this:

"Even highly radio-active solid wastes can be disposed of safely in the sea provided all relevant factors are kept in mind: movement of the surface water, the breeding and migratory habits of fish, and the possible hazard to seaweed where it is harvested for food, fertilization, or industrial use." [2]

Perhaps, doctor, you can see why I grew alarmed.  I knew little of food chains and concentrations and nothing of the effects of ionising radiation on tissues.  I'd barely heard of the concept of half-life, but these predictions just felt wrong. They seemed like a bad future.

Being a compulsive book gatherer, I started collecting old books which offered visions of the future.  I found they always warned us to make way for the bright new future, to plan, to prepare, and to act now.  I also noticed that the future always included flying cars and a world where we had swapped eating for the efficiency and superior nutritional value of food pills.

What's that, doctor?  No, I don't want to take two fish and chips pills and call you in the morning. May I continue?  Thank you.

Popular Science, 1930s.
Popular Science, 1930s.
Another common early 20th century theme was the modern airliner. This had all the frills of a 1920s luxury ocean liner, but slung beneath the gas bag of a motorised balloon.  These Zeppelins on the grand scale each had a ballroom, a dining room seating 129 fashionably attired gentlemen and ladies, with space for dozens of servants and individual cabins, each fitted with a bath and a shower. Modern cattle class, eat your heart out!

And one more definite, absolutely certain future: the pneumatic telegraph.  This did in fact happen in a small way.  It was still around when I was old enough to be aware of it at a local clothing shop, where money and sales dockets were carried through tubes to a cashier on another floor, and the change and a receipt were returned similarly.  It was scaled-down, but still clearly the pneumatic telegraph.
From Scientific American, 1860s.

The money and paperwork travelled in a small cylinder, just the right size for the tubes.  It was neat and effective, but a mere shadow of the projected pneumatic telegraph, which 19th century seers thought would replace messenger boys and telegraphy, carrying messages and parcels unerringly to their destinations though tubes, using high pressure behind or low pressure in front—or both.
Readers were assured that these innovations would revolutionise their lives.  Capitalists were admonished in the early 1860s to put money into these schemes, ignoring all others.

Now imagine a world where farmers had walked off the land, because, let's face it, food pills would soon do them out of the job.  Think of a world where the roads had been dug up and sold off to developers, because flying cars need no roads.

Contemplate a world where the cables have all been torn down or dug up and sold for scrap, because the pneumatic telegraph was coming.  Ponder a world where the airports have been sold off to robber barons and covered with towering future slums, intended to house the otherwise unemployed farmers, road-makers and cable layers, because modern airships don't need long runways.

Nobody would plan on that basis, would they, doctor?  I mean, it's ludicrous, isn't it?  What sort of idiot would plan for the future, based on half-baked notions, pipe-dreams?  Who would base decisions on special pleading from interested parties, all eager to hack out an empire for themselves?
So, why do people happily embrace the prospect of a world without libraries, based on the prediction that we don't need books or libraries any more, because we can get everything we need from the internet?

Those who make sweeping assertions like this don't know what books are, have no sense of what libraries do, and absolutely no idea of what the internet is—or offers.  Most importantly, these rigid descendants of Wackford Squeers lack the wit to see that institutions evolve.  Change is attained through finesse, using scalpels, files and sandpaper, not bulldozers, flamethrowers and explosives!

OK, doctor, so I'm probably biased, being a book collector and writer who likes hanging out with librarians, but at least I've studied change and how it happens.  Perhaps that doesn't qualify me to predict the future, but it gives me a clear insight into what doesn't work.

One profound and simple change to our society came when a bright spark had the idea of combining the mouse and the graphic user interface to make it easier for people to access their computers.  Most people credit Alan Kay with this idea, and he was certainly involved in creating our mouse and windows world.  That's why I like his comment:

"Don't worry about what anybody else is going to do… The best way to predict the future is to invent it." [3]

He said that in 1971, forty years ago, probably before the mouse and GUI came up.  But then he did the necessary work.  The people like that who do things are the ones that make change happen.  Some may make better mousetraps or a brighter light, others just nudge the system. These are small moves that by themselves do little.  Together, they make a Brave New World.

One emerging cause of change is collaboration.  For all the problems caused by mischievous people, Wikipedia is an amazing resource of ideas, notes and reminders.  As more and more people offer their expertise, adding original sources and detail, it continues to outgrow the silly games of the vandals.
You can see one of my contributions in the entry on the Flint Piano or lithophone.  I made it after I came across some original material, looked at Wikipedia for more detail, found the account deficient, and rewrote the article, using what I'd found.

Another time, I added the text (and a pointer to the source) of a first-hand account of an 1811 earthquake on the New Madrid fault in America.  That tremor made the Mississippi River run backwards, and it rang church bells in Boston.  I tripped over the account while seeking the origins of the word 'diggings', and thought it interesting.  At some time in the not-too-distant future, that fault will move again.  When it does, my contributions will be there, waiting and accessible.

The grand planners and self-promoters want to scrap everything, so only their vision of the future can be enacted.  Sounds a bit like China's Cultural Revolution, doesn't it?  I'd rather see cultural evolution, where we get to keep all the best bits, and preserve the spare bits for a while at least, in a box in the garage.

In 1990, two mathematicians, Philip Davis and Reuben Hersh, offered a refreshing view of computer art. They thought its future lay in 

"… the dynamic, the animated, the interactive.  It should look not towards Rembrandt, but towards Verdi's 'Aïda'.  Not just the classical 'Aïda', but an 'Aïda' with the audience singing along and scrambling onto the backs of the elephants on stage.  Chaos?  No.  Total theatre." [4]

This is the way our society is quietly heading.  Invisibly, it's happening, all around us.  The National Library of Australia (who are, let me confess, sometimes my publisher) have a remarkable resource called Trove.  This is a massive collection of digitised newspapers, linked to computer-generated text which registered users can correct.

More than that, they can add generic tags like "bushrangers", "39th Battalion", or one of my favourites, "early use of language", the tag I hang on the earliest instances I find of words like 'squatter', 'billy', 'bludger', 'swag' and 'fossick'—among others.  Those tags are there for all time.
Equally, I was researching two London conmen called Tripe and Montague, who in 1852, fleeced would-be emigrants who were headed for Australia's goldfields.

I found their trial transcript at the Old Bailey, but until I found a record of their pardon, I believed they had been transported to Australia.  I grinned at their probable reception here, if they met any of those they had robbed.

Curious, I burrowed and found an 1856 classified ad in the Sydney Morning Herald where 'W.W.' was advertising for friends who'd been caught up in the swindle. Trawling along, I found another ad in the same paper in 1865, where William Waterford was mentioned. I went back to the first ad, and attached a comment, drawing any later reader's attention to the second ad and the likelihood that the mysterious W. W. was in fact William Waterford.

Slowly, the resource is growing, and the prospects for future researchers are being quietly enhanced, not by hewing and slashing behemoths, but by nibbling and gnawing mice.

One day, I hope, people will turn around, raise their eyebrows and ask "Where did all that come from?" That day isn't here yet, but we can hope for it, and work towards it. If I'm wrong, no harm will've been done, because nothing is scrapped as these small additions drop into place.

But we're not there yet. The other day, I was looking at an old Times Literary Supplement in the State Library of New South Wales' online collections, when another article on the same page caught my eye.  It mentioned Sir Horace Mann appearing in a painting in Florence.  I soon found Zoffany's The Tribuna of the Uffizi and with some effort, tracked down a key to the people in the painting.

I wanted to annotate the TLS article so others could find the same resource, but I couldn't.  One day, that'll be a given, because every system will have that sort of flexibility worked into it.  As a stopgap, I edited the relevant entry in Wikipedia, and added a link there.

That's the future I look forward to.  The difference between my vision and that of the people who'd sack the librarians, pulp the books and demolish the shelves, all to make room for soft lounges, a couple of computers and a gleaming great coffee machine is this: my vision offers hope, and doesn't prescribe world-crashing and irreversible destruction.  We build on what we have, knowing that some of the old stuff—and some of the new—will be discarded, once we're in a position to decide.

As the twig bends, so the tree bends.  A future built on collaboration relies on people who gain a quiet joy from contributing gems, nuggets and crumbs to future generations, whimsical folk who amuse themselves by committing acts of anonymous scholarship.

You see, doctor, this is the way the world bends—not with a bang, but with a whimsy.


[1] Egon Larsen, Atomic Energy, Pan Books, 1958, p. 136-7.
[2] Egon Larsen, Atomic Energy, Pan Books, 1958, p. 136.
[3] http://www.smalltalk.org/alankay.html
[4] Philip J. Davis and Reuben Hersh, Descartes' Dream, Penguin, 1990, p. 53.

Sunday, 26 April 2015

Crooked Mick fails to fight

I never got around to mentioning this, but Crooked Mick came down to the Big Smoke once.  He wasn't there long, though, before he got himself into trouble.  There was this bloke in town, claimed to be the world heavyweight champion of the world.  Well as you might guess, with two big men in the same town, the inevitable happened, and Mick wandered into a room in a fancy hotel, where this champion was holding court and showing off.  Mick just stood there quietly, watching while the champion bent a steel bar into a U-shape.  It was quite a thick bar, and the champion made quite a fuss about how hard the job was.

Then to prove how strong he was, he bent two more, even thicker, bars in the same way, and passed them around.  Two men took hold of the thinnest bar and tried to straighten it again, but they couldn't.  Then while everybody was still crowding around to look at the bars, the bloke moved over to two huge iron barbells, and a pile of spare weights.  Calling for their attention, he lifted first one weight, and then the other.  He was about to start adding extra weights, but he stopped, because there was a fuss going on in the background, and nobody seemed to be paying him much attention.

What had happened was that Crooked Mick had been looking at the three iron bars on the floor, and being a tidy sort of bloke, he picked them up to put them on a table.  Before he put them down, and without really thinking about it, he had bent them all straight again.  That would have been acceptable, but Mick had straightened all three at the one time, and the heavyweight champion of the world got really mad about this.  You could see he was really seething with anger, under the surface, and he was muttering something real nasty.

Anyhow, he tried not to let it show that he was angry, but I was there, and he said, all sarcastic like, "I suppose you'd like to try lifting these weights as well?".

Well Mick said nothing but walked over, and looked at the two barbells on the floor.  Everybody crowded around, and Mick looked at the weights, then says, "Most of you couldn't see me here, so hang on, and I'll move over near the door."

And with that, he picks up one barbell, tucks it under his arm, then slips his little finger around the second one, getting the balance just right and then scoops up all but one of the spare weights in his other hand, and starts to saunter across to a better spot.  "I'll come back for the other one," he says, "it's better not to strain yourself with stuff like this!"

Now I can see what you must be thinking: them weights were fakes, but it wasn't so: they were the real thing all right, because when he stopped in one place, the floor collapsed under Crooked Mick.  He plummeted down three floors, we found out later, but before we could all race out to see what had happened to him, he comes back up the stairs, still carrying the weights and the barbells.

Well that was too much for the champion.  He grabs a white glove from his pocket and steps forward, slapping Mick on the face with it.  "We will fight a duel!" he shouts.  "You will choose the weapons!"

Flash Jack was there, and so he takes Mick to one side and explains to him about the etiquette of duelling, and how the challenged person has choice of weapons, and how you always fight at dawn.  Crooked Mick stood there thinking for a moment, then says, "Righto.  I choose to fight with axes."

"I don't have an axe!" says the champion, starting to look worried.

"No worries, sport!" says Mick.  "I use two most of the time, so I'll bring 'em both, one for each of us.  See yer tomorrow!"

We never did though, even though we turned up at the park on time, with the axes.  In fact, we never saw the heavyweight champion of the world at all after that, and Mick went back to the Speewah the very next day, mumbling something about city folk being too rough for the likes of him.

In spite of what some people say, he was a very gentle bloke at heart, and I don't know what he would've done if the heavyweight champion of the world had tried to take him on.  As far as I know, Mick only ever fought one fight, and he lost that.

So when you hear people saying Mick'd rather have a fight than a feed, I reckon they're having a lend of you.


* * * * *

Note: there is a whole book of these stories, which I have more or less given up on pitching to publishers, so they will probably appear in an e-book.

There will be quite a number of these on the blog, all with the tags Speewah and Crooked Mick.