seek echo

Hyperbolic Crochet Coral Reef

2009-04-21T05:28:00.001-07:00

IFF Hyperbolic Crochet Coral Reef
Originally uploaded by Margaret Wertheim.

I watched the TED talk given by Margaret Wertheim with great pleasure. Here is a project that links to so many interests, the mathematical, the playful, the environmental, the practical, the educational. It connects them together with beauty and surprise.
These beautiful crocheted corals are part of a hugh ("viral") project to model a coral reef in wool. They model hyperbolic space by the clever crochet trick of regularly adding number of stitches to each row. And this works because corals have hyperbolic shapes.

She describes mathematicians as the "free-est of thinkers", but they never noticed that these-hard-to-model hyperbolic planes were in front of them on their salad plate. She also mentions the work of Froebel in inventing Kindergarten. Have a look at the twenty gifts, through which he instilled mathematical thinking through physical play.

There's a lot of really good stuff on the Institute for Figuring's site, for instance the explanation of hyperbolic space.

In the interview with her on TED, she describes her own school learning and her discovery of pi:

I think that my love of figures and figuring is a thing that's bound up with my childhood. When I was in grade three or four, my mathematics teacher, a man named Mr. Marshall, gave us a mathematics lesson about circles. The whole point of this lesson was to teach us about pi; the magical number that is at the heart of all circles.
Instead of simply telling us the formula for the circumference of a circle and the area of a circle, he gave us an entire lesson letting us discover pi for ourselves. For me, the exercise worked. I looked around me and I realized that every time I see a dinner plate, every time I see the sun or the moon, every time I see the wheel of a car, every time I see a circle in the world around me, that this magical number pi is embedded in it.
I will never forget this moment; it was truly like a revelation to me, that this almost angelic thing pi was hovering magically, like an angel behind the material world. I had several experiences like that during my childhood.

This all seems to link back to the beginnings of maths in Europe with Thales, Pythagoras and co. They had a way, ways, of figuring - using arrays of pebbles to explore number theory, and the straight line and edge to explore geometry. There were limitations to these ways of modelling, but even the limitations were creative. They created a ludic quality - as in recreational maths - "what spaces can I get to within the constraints of this structure". Like origami, certain things are possible, certain things are not possible. And as "ontogeny recapitulates phylogeny"so education can recapitulate history; just as maths was invented or discovered through these games, so in the present it can be discovered anew by children or adults. Time to get crocheting...

♓

2009-04-03T13:18:00.000-07:00

A cord joins the tails of Pisces, the two fishes. From the Atlas Coelestis of John Flamsteed.

It's never interested me before, my star-sign, Pisces. But there is, naturally, some kind of story behind it...

See for instance this page...

"The mythological events concerning this constellation are said to have taken place around the Euphrates river, a strong indication that the Greeks inherited this constellation from the Babylonians. The story follows an early episode in Greek mythology, in which the gods of Olympus had defeated the Titans and the Giants in a power struggle. Mother Earth, also known as Gaia, had another nasty surprise in store for the gods. She coupled with Tartarus, the lowest region of the Underworld where Zeus had imprisoned the Titans, and from this unlikely union came Typhon, the most awful monster the world had ever seen.

According to Hesiod, Typhon had a hundred dragon’s heads from which black tongues flicked out. Fire blazed from the eyes in each of these heads, and from them came a cacophony of sound: sometimes ethereal voices which gods could understand, while at other times Typhon bellowed like a bull, roared like a lion, yelped like puppies or hissed like a nest of snakes.

Gaia sent this fearsome monster to attack the gods. Pan saw him coming and alerted the others with a shout. Pan himself jumped into the river and changed his form into a goat-fish, represented by the constellation Capricornus, also inherited from the Babylonians.

Aphrodite and her son Eros took cover among the reeds on the banks of the Euphrates, but when the wind rustled the undergrowth Aphrodite became fearful. Holding Eros in her lap she called for help to the water nymphs and leapt into the river. In one version of the story, two fishes swam up and carried Aphrodite and Eros to safety on their backs, although in another version the two refugees were themselves changed into fish. The mythologists said that because of this story the Syrians would not eat fish. An alternative story, given by Hyginus in the Fabulae, is that an egg fell into the Euphrates and was rolled to the shore by some fish. Doves sat on the egg and from it hatched Aphrodite who, in gratitude, put the fish in the sky. Eratosthenes wrote that the two fishes represented by Pisces were offspring of the fish that is represented by the constellation Piscis Austrinus.

In the sky, the two fish of Pisces are represented swimming in opposite directions, their tails joined by a cord. The Greeks offered no good explanation for this cord, but according to the historian Paul Kunitzsch the Babylonians visualized a pair of fish joined by a cord in this area, so evidently the Greeks borrowed this idea although the significance of the cord was lost.

Pisces is a disappointingly faint constellation, its brightest stars being of only fourth magnitude. Alpha Piscium is called Alrescha, from the Arabic name meaning ‘the cord’. It lies where the cords joining the two fish are knotted together. Pisces is notable because it contains the point at which the Sun crosses the celestial equator into the northern hemisphere each year. This point, called the vernal equinox, originally lay in Aries but it has now moved into Pisces because of a slow wobble of the Earth on its axis called precession."

Could it mean anything, Aphrodite and Eros under threat, swimming in opposite directions, but held together with a cord to escape together?

It seems full of meaning, but am I just projecting my own meaning onto the myth??

Maybe it's time to read about the other star signs...

♈ ♉ ♊ ♋ ♌ ♍ ♎ ♏ ♐ ♑ ♒ ♓

money and knowledge

2009-03-10T01:32:00.000-07:00

Some people see it as no coincidence that the 'Greek Miracle", the explosion of science, maths, arts from the 6th Century BC onwards, coincided with the beginnings of standardised currency in gold-rich asia minor.

But though it may, along with the network of coastal Greek colonies linked to the Ionian city states, must have contributed to the material prosperity of the times (especially if you ruled a pirate state), and allowed surplus wealth to be dedicated to the arts and learning, the new kind of money must have brought its own worries. And of course wealth and increased individual choice had existed before, with its own problems, which continue to this day.

aristotle

2009-02-26T13:34:00.000-08:00

Picking up Aristotle is not always entertaining. Apparently, according to Wikipedia, all his more entertaining pieces were lost, although the encyclopedia does acknowledge, "a few modern scholars have actually admired the concise writing style found in Aristotle's extant works."

But these words of his about the earliest scientists (he means Thales & co.) caught my eye:

"It is owing to their wonder that men both now begin and at first began to philosophize; they wondered origanally at the obvious difficulties, then advanced little by little and stated difficutlties about the greatest mattres, e.g. about the phenomena of the moon and those of the sun and of the stars and about the genesis of the universe. And a man who is puzzled and wonders thinks himself ignorant (whence even the lover of myth is in a sense a lover of Wisdom, for myth is composed of wonders)..."

(Notice that even Aristotle knows that myth is not just bad science, pre-science, as some people imagine. It is not, even if from time to time it appears to be, really trying to explain the sun and the moon and their movements. That story about the Tortoise and the Geese might seem like it's trying to explain how the tortoise has a cracked-looking shell, but that's not what it's about at all.)

So, leaving aside that question of "greatest matters", Science, Myth are the same in this, they are both about, or at least begin with, Wonder and Puzzlement.

'Why do those isands appear above the horizon when we climb the mountain? Why are the stars different when we sail north or south? ...'

And the maker of myth, the poet, the storyteller, makes his myth because he wants his hearer to wonder and puzzle about human life.

Take... at random, take the tale of Narcissus and Echo. Strange story. What's it on about? That's the first question. People talk about narcissism - so it's about a distorted self-regard? Perhaps, but why stop wondering there? What else is it about? Echo is in the tale too, what does she represent?

The myths, the stories are made to be many-faced. If they weren't, they wouldn't cause any puzzlement, any wonder. There is not meant to be a moral at the end of the fable. Or if there is, it should be a red herring, something that makes the story more puzzling still.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Narcissus, the beautiful young man who one day parted some reeds and looked into a still pool and when he saw his reflection fell in love with it. He went back again and again to stare into the pool, into his own reflection. He was too much in love with it to even notice that Echo loved him. She spoke to him, but he did not hear, did not reply and so she just faded away. Even her voice faded away, until all it could do was repeat a little of what was said. “Oh, how I love me!” Narcissus said. “Love me!” Echo replied. “”You, in the water, be mine!” Narcissus said. “Be mine!” was Echo’s answer. Narcissus of course became the yellow flower beside the pool. And Echo – well, she is Echo.

Heraclitus

2009-02-03T02:22:00.000-08:00

Among all the much later writings that we rely on for our picture of Pythagoras there are often fragments of earlier authors that seem to go back, sometimes to the actual time of Pythagoras. Examples: a few of the sayings of his contemporary, Heraclitus:

-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:

"Much learning does not teach understanding, otherwise it would have taught Hesiod and Pythagoras, Xenophanes and Hecataeus."

"Pythagoras, the son of Mnesarchus, practiced inquiry most of all men and having made a selection from these writings made for himself a wisdom, a polymathy, an evil trickery.”

-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:

He gives the appearance of having a low estimation of Pythagoras, but in passing gives us an impression of him:

having "much learning"
practicing "inquiry" more than anyone else,
using the writings of others,
developing a "wisdom"
being a polymath

These things, almost more than any "content" to the learning, interest me most. I don't think it was "academic" learning that he had. A better picture might come from the reported sayings of the "Seven Sages". Their "learning" was more concerned with - what can we call it? - - how to live well?

-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:

ancient computer

2008-12-21T08:24:00.000-08:00

I blogged this device a long time ago

The accompanying New Scientist article discusses how people have seen a new angle to this:

"MARCELLUS and his men blockaded Syracuse, in Sicily, for two years. The Roman general expected to conquer the Greek city state easily, but the ingenious siege towers and catapults designed by Archimedes helped to keep his troops at bay.
Then, in 212 BC, the Syracusans neglected their defences during a festival to the goddess Artemis, and the Romans finally breached the city walls. Marcellus wanted Archimedes alive, but it wasn't to be. According to ancient historians, Archimedes was killed in the chaos; by one account a soldier ran him through with a sword as he was in the middle of a mathematical proof.
One of Archimedes's creations was saved, though. The general took back to Rome a mechanical bronze sphere that showed the motions of the sun, moon and planets as seen from Earth.
Special admiration
The sphere stayed in Marcellus's family for generations, until the Roman author Cicero saw it in the first century BC. "The invention of Archimedes deserves special admiration because he had thought out a way to represent accurately by a single device for turning the globe those various and divergent movements with their different rates of speed," he wrote. "The moon was always as many revolutions behind the sun on the bronze contrivance as would agree with the number of days it was behind it in the sky."
Until recently, historians paid scant attention to this story: the description suggests a sophisticated mechanical device, beyond anything the ancient Greeks were thought to have been capable of. Furthermore, Cicero had no technical training, and did not explain how the device worked. He could have made the story up for effect.
Now, however, research on the battered remains of a mysterious ancient device suggests that Cicero was telling the truth. While the Antikythera mechanism is not the same one seen by Cicero - it was not made until a century later - it proves that clockwork mechanisms like the one he described really did exist, and that ancient Greek technology was far more advanced than thought. Freshly deciphered inscriptions on its dials also hint at the origins of this technology..."

and another thing

2008-11-23T14:13:00.000-08:00

saying of the Pythagoreans:

"Two-footed is a human being, and a bird, and a third thing as well."

irrational

2008-10-30T15:47:00.000-07:00

The story of the discovery of irrational numbers is shrouded in mystery, and linked with a certain early Pythagorean named Hippasus. The popular version has rumours of cover-up, betrayal, perhaps even murder.

I have taken the proof of their existence, and, seeing that the Socratic dialogue is still a contemporary form, re-created a later conversation between Socrates and the slave boy we meet in Meno.

Socrates: So you’ve kept going with your mathematical studies?
Boy: Yes, you made me see that it was possible, Socrates. I have, as you said, approached the same questions in many different ways, and have indeed gone far beyond what we talked about last time.
Socrates: And what in particular have you been studying?
Boy: Too many things to say.
Socrates: Tell me at least one or two of the areas.
Boy: Where can I begin? Factors, square numbers, whole numbers, fractions…
Socrates: So, many of the kinds of numbers…
Boy: You could say all of the numbers, as all numbers are either whole numbers, or a fraction of some kind.
Socrates: Well that’s an interesting suggestion. Are you sure you’ve included all possible numbers?
Boy: How could there be any other?
Socrates: Well, you remember that diagonal we talked about last time?
Boy: When we were doubling the size of the square, yes.
Socrates: I think I can prove to you that the side of a square and its diagonal could not possibly both be whole numbers or fractions.
Boy: How could that be?
Socrates: Well let’s draw that same diagram again.

Socrates: Let’s propose that the side and diagonal of the small square are both whole numbers, the smallest possible whole numbers they could be. Do you understand what I am proposing?
Boy: Yes, I do.
Socrates: So they could not both be, let’s say, even, could they?
Boy: Obviously not. You would be able to halve both numbers and still have whole numbers, and you proposed that they were the smallest possible whole numbers.
Socrates: I see that you have indeed, as you say, been studying. Now, if the side of the small square is a whole number, you will agree that the square itself must be whole a whole number.
Boy: Yes, that’s obvious.
Socrates: Forgive me if I move in steps that are too obvious. I only want to be sure that we have not taken some wrong turning. Now we saw last time, did we not, that the square on the diagonal is twice the size of the small square?
Boy: Yes, how could I forget!
Socrates: Then that square on the diagonal, as it is double a whole number, must be an even number?
Boy: Agreed.
Socrates: And would you agree too that if the square is even, then the side of the square must be even too?
Boy: I’ll need to think about that a moment, let me see…

Boy: Yes, I can see now that all even squares will have an even side.
Socrates: So, we have established that the diagonal must be an even number.
But the way you’ve drawn those squares, shows something else, did you see? – They are always multiples of four too?
Boy: Yes…
Socrates: And the little square is half the size of the square on the diagonal, so it must, you’ll agree, be half a multiple of four?
Boy. Yes… which will be a multiple of two.
Socrates: I can see it’s not so easy for me to race ahead of you anymore. Yes, it will be an even number. And even squares have even sides as we said, so the side of the small square is even too.
Boy: Yes.
Socrates: But don’t you see, there is something absurd here? We started by saying the side and the diagonal would be the smallest possible numbers. But now we find that they are both even.
Boy: Yes, which means they could both be divided by two to get smaller whole numbers. How can that be?
Socrates: How indeed? Are you sure that all the steps we have taken are acceptable? I haven’t moved you on too quickly, making you agree with something that is uncertain by sheer force of personality, have I?
Boy: No, Socrates, each step was clear enough. And yet we arrive at an absurdity…?
Socrates: The only way out of it that I can see – tell me if you can see another – is that the proposition we started with, that there are two smallest whole numbers that could measure the side and diagonal of the square is an impossibility: there are no such numbers.
Boy: But that seems illogical. These lines must after all have a particular length?
Socrates: That I do not deny. But both of them cannot be whole numbers – or, as it is simple enough to demonstrate, fractions for that matter.
Boy: Is there not something unreasonable about that proposition, that there are measurements which cannot be expressed either as whole numbers or as fractions?
Socrates: We cannot exactly call it a proposition, since it is something more certain that we have arrived at by carefully following the trail left by the numbers themselves. But I know what you mean about it’s seeming unreasonableness, and with your permission I propose to call such numbers “irrational numbers”.

not so ideal

2008-10-28T01:15:00.001-07:00

I really shouldn't carp at any supposed shortcomings in Socrates' method of questioning.
Any real attempt to do the same, even tidied up, with interruptions removed, would be something more like this real conversation...

Socrates: Could you make a square twice the area of that square?

Boy: No.

Socrates: What sort of square could you make then?

Boy: You could make a rectangle.

Socrates: OK, so there’s one square, there’ another, there’s a third, then a fourth. You could also draw lines from corner to corner like that…

Boy: I also thought that maybe if we squished that up, if I squished them together, we could make the rectangle into a square. Like play-doh™.

Socrates: So you’d keep the same area, and push in the short side of the rectangle?

Boy: Yes.

Socrates: OK, so there’s a line from that corner to that corner… and do you see those triangles now?

Boy: Yes. There, there, there… there’s eight. Because I know that there’s two here, so that must be four, so that must be eight.

Socrates: Good. Are all these triangles the same size?

Boy: Yes.

Socrates: Do you see this square in the middle?

Boy: Yes. It’s a diamond. Well, it’s not a diamond. Because that is a square. And a diamond… Say this (draws) was a square, even though it isn’t, and then we do that (rotates)… that wouldn’t be a diamond. But this (draws) … that would be a diamond.

Socrates: A diamond has to be longer one way than the other?

Boy: Yes

Socrates: Right. So that’s a square.

Boy: That’s a square on its point.

Socrates: How many triangles were in that first square I showed you?

Boy: Two.

Socrates: And this last square on its point has…?

Boy: Four.

Socrates: So what can we say about the squares, their two sizes?

Boy: Er… they’re the same size…

Socrates: What did I ask you to do in the beginning?

Boy: You asked me to…

Socrates: Make a square that was double. So the first square has how many triangles?

Boy: Two.

Socrates: And the last one?

Boy: Four.

Socrates: Two... and four?

Boy: …

Socrates: So isn’t that middle square double the first one?

Boy: Oh yeah!

doubling a square

2008-10-26T08:29:00.000-07:00

Looking at that famous part in Plato's Meno:

SOCRATES: Tell me, boy, is not this our square of four feet? You understand?

BOY: Yes.
SOCRATES: Now we can add another equal to it like this?

BOY: Yes.
SOCRATES: And a third here, equal toeach of the others?

BOY: Yes.
SOCRATES: And then we can fill in this one in the comer?

BOY: Yes.
SOCRATES: Then here we have four equal squares?
BOY: Yes.

SOCRATES: And how many times the size of the first square is the whole?
BOY: Four times.
SOCRATES: And we want one double the size. You remember?
BOY: Yes.
SOCRATES: Now does this line going from comer to comer cut each of these squares in half?

BOY: Yes.
SOCRATES: And these are four equal lines enclosing this area?

BOY: They are.
SOCRATES: Now think. How big is this area?
BOY: I don't understand.
SOCRATES: Here are four squares. Has not each line cut off the inner half of each of them?
BOY: Yes.
SOCRATES: And how many such halves are there in this figure?

BOY: Four.
SOCRATES: And how many in this one?

BOY: Two.
SOCRATES: And what is the relation of four to two?
BOY: Double.
SOCRATES: How big is this figure then?

BOY: Eight feet.
SOCRATES: On what base?
BOY: This one.

SOCRATES: The line which goes from comer to comer of the square of four feet?
BOY: Yes.
SOCRATES: The technical name for it is 'diagonal'; so if we use that name, it is your personal opinion that the square on the diagonal of the original square is double its area.
BOY: That is so, Socrates.
SOCRATES: What do you think, Meno? Has he answered with any opinions that were not his own?
MENO: No, they were all his.
SOCRATES: Yet he did not know, as we agreed a few minutes ago.
MENO: True.
SOCRATES: But these opinions were somewhere in him, were they not?
MENO: Yes.
SOCRATES: So a man who does not know has in himself true opinions on a subject without having knowledge.
MENO: It would appear so.
SOCRATES: At present these opinions, being newly aroused, have a dream-like quality. But if the same questions are put to him on many occasions and in different ways, you can see that in the end he will have a knowledge on the subject as accurate as anybody's.
MENO: Probably.
SOCRATES: This knowledge will not come from teaching but from questioning. He will recover it for himself.
MENO: Yes.

Not being a mathematician, I like the simplicity of this demonstration.

And I also like it that way back then, this careful, step-by-step approach was being used. In "hindsight" I think the steps could have been a little different, a little more time taken to talk round the figure, to avoid the "I don't understand".

But Socrates makes up for any shortcomings in his questioning (!) by acknowledging the need for "many different ways" and "many different occasions".

evidence

2008-10-13T09:01:00.001-07:00

The Zones
Originally uploaded by jovike.

I touched on the question of empirical evidence before. And these were empirical people - they - Thales, Pherecydes, Anaximander - were busy with sundials, and day lengths and the like.

I must have mentioned this abstract before:

"Abstract. Probably the most direct observation of the Earth’s curvature is how objects appear from over the horizon when we approach them and disappear as we get further away from them. Similarly, the portion of a high object (a building or a mountain) that is visible depends on the height of the site where the observation is made. Based upon these very obvious facts, a simple method to estimate the Earth's radius R has been applied. The method does not need either sophisticated instrumentation or complex mathematics. In our application of the method presented here, the result is R = 6600 ± 600 km in the best case. A discussion is presented about the possible use of this method in ancient times. Surprisingly enough, we have not found any reference to the use of this method despite its being simpler than, for example, the classical approach of Eratosthenes."

And other evidence??

Here's a list I put together when I started thinking about this:

height of stars, sun, moon, planets varies with latitude
ships disappearing at a set distance from port
the moon is a sphere
the sphere of the sky rolls ‘under’ the earth (support impossible)
eclipse
sun on mountain?
Islands missing their bottom half
More islands becoming visible as one climbs
Sunrise and sunset at different longitudes
Apparent curvature of earth up high mountains
Mountains appearing shorter than they ought to be
The pattern of day length through the year
Time zones (?!)
Variation in day length through year?
The clouds, light on them at dawn and dusk, the way they appear at the horizon

Some of them I'm not so sure of, some are glaringly obvious. To us, now - and would have been for whoever looked then, and many people did.

explanatory power

2008-10-13T08:16:00.000-07:00

Wikipedia says that Pythagoras "saw harmony in the universe and sought to explain it. He reasoned that Earth and the other planets must be spheres, since the most harmonious geometric solid form is a sphere."

But is there really any evidence that this was his reasoning? Why shouldn't everything be spherical if this is so perfect?

Aside from all the observational reasons he could have had for thinking the Earth is a sphere, there is the sheer explanatory power.

Coldness in the north is explained, and hotness in the south. The height of the sun and the stars changing with latitiude.

And if you tilt the axis of the Earth (or for that matter of the Heavens) you explain even more - the seasons, temperature changes, day length varying from north to south and through the years, the skies changing through the year.

You'd have to play with a sphere for a while to see all this.

Corvus

2008-09-14T15:14:00.001-07:00

He asked us what constellations we could see and we named them. When we came to the Raven and the Snake, Rhoda told us that she used to sing the song of the Raven and the Snake, called ‘Apollo’s Command’.

Pythagoras got up and gave her the lyre.

‘Sing Apollo’s Command for us now.’

‘I can’t,’ she said, trying to give the lyre back.

‘But you must. Sing it,’ he insisted and she took the lyre from him and began to pick the strings. She sang out to the stars:

‘Apollo prepared a feast for Zeus,

Sent Raven down to earth with a golden cup

“Fetch me some fresh spring water,” he said,

Raven went down, found the spring

Landed on a fig tree.

Looked at the figs, hard and green.

Forgot the cup, forgot the spring,

Thought only of the figs on that tree.

And waited, waited,

Waited till they were soft and sweet,

And with ripe figs filled his belly.

Then Raven saw the golden cup,

Remembered Apollo’s command,

Picked up the cup and filled it up,

Wondering what to say when he saw the god.

Just then Snake shot over the pool;

Raven saw his chance and grabbed it,

And with water and Snake flew back to the god.

“You’ve been gone so long, what kept you?”

Raven dropped down the snake.

“It was this evil thing that was guarding the spring

I told him he would be punished.”

Apollo, who knows truth from lies,

Said “I see your belly, Raven.

It’s full of the figs you waited for.

It’s you, not Snake, I’ll punish.

From now on your cry will always be rasping and dry;

No spring will ever quench your thirst.”

And to remember this

Apollo took Raven, cup and Snake

Reached up to the night sky

And hung them for all to see.’

Rhoda finished, stood up and gave the lyre back to Pythagoras.

‘Now I know why Aesop sent you to me,’ he said. ‘It was so that we could sit up here, high up on Mount Kerkis and hear you sing. Do you know more like that?’

donkeys and sitting

2008-09-11T14:00:00.001-07:00

Still on the question of 'why maths', why in ancient history or prehistory did someone start it up? Why did the 'Milesian Philosophers' go in for it? ...

There is the story of Nasrudin and the donkeys.

His friend had asked him to bring nine of them back from the market. He found that when he was riding on them there were eight. But when he got off there were nine.

(The picture is to be found here, along with a fuller telling of the tale.)

When Nasrudin got seated on a donkey, the counting went wrong. When he did the counting properly, he wasn't sitting on a donkey.

So the link? Walking and counting go together. Sitting on donkeys goes with getting the wrong answer.

So, maths as a way of getting a man off his donkey?

Speakers' corner

2008-07-18T15:42:00.001-07:00

Speaker's corner ~ Silently waiting for an audience
Originally uploaded by Boreio Selas.

When I was just a teenager Paul used to speak to big audiences at Speakers' Corner. Now, in the noisy show of the place he just, like Diogenes with his lamp, stands and waits for an individual human being.

A story retold from back then:

A girl who had been in London for some time, from Germany -
She was asked "Where is Virgin Records?"
She pointed the way.
"Where is MacDonald's?"
She pointed it out.
- "Oxford Street?" - "Picadilly?" - she knew all these.
Then she was asked, "And where can I find wisdom?"
She got out her dictionary.
Paul said "You've been wasting your time. It's things like Wisdom, Courage, Beauty that makes you human, not these other things. You've been here all this time - and you haven't learnt the essential things."

§§§§§

It is related that when Socrates first met Xenophon, then a boy, in the street he stopping him and asked where various articles could be got.

Xenophon told him. Socrates then asked, ‘Where can you get brave and virtuous men?’ and when Xenophon was puzzled told him to come with him.

connections

2008-06-12T02:18:00.001-07:00

the tube
Originally uploaded by simonsterg.

As a kid I got to know various bits of London. Often I got to them by Tube. Starting from my home in Paddington, I popped up in South Kensington or Tower Hill or Baker Street. They seemed like very separate places, unconnected, with their own climates and moods. Only later when I moved about overground a lot more did the islands of familiarity join up into one London.

I'm not a mathematician, but you can get a sense of how the world of number interconnects.

Take this picture from two posts back:

I thought "what if I tidy this up a bit? - make a nice neat diagonal across the rectangles - what then?" Like this:

This is an intriguing shape...

The triangles, with sides 3,4,5 (the first and most familiar "pythagorean triple") have an area of 6 squares.

The diagonal square in the middle is a 5 by 5 square and has an area of 25 squares.

And - yes -it connects with this proof of The Theorem - See!

"See!"

"don't wear a god's image on your ring"

2008-06-03T14:26:00.000-07:00

I'm going into windbag territory here... I've got my special ring on... If I don't come out in a few days, come and get me...

Academics naturally are tuned into ideas. They want to read what people believed, what their "philosophy" was.
But like Borges they / we could be missing everything.

Wikipedia has a very typical derivation for the word "philosophy":

The word philosophy is of Ancient Greek origin: φιλοσοφία (philosophía), meaning "love of knowledge", "love of wisdom".

The Pythagoras page says that:

He was the first man to call himself a philosopher, or lover of wisdom.

But going back to what Diogenes Laertius said:

The first... to call himself a philosopher (lover, friend of wisdom) was Pythagoras. "For no one," he said, "is wise (sophos) except god."

(Lives of the Philosophers, i. 12)

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

He wanted to contrast himself with "the wise", the sophists, and adopt a name that was more modest. He called himself a (mere) friend of wisdom.

Still, even to be the cousin, or acquaintance, or next door neighbour of wisdom would be a fine thing.

Wisdom takes the study of the ancients out of the academic realm. We all know, even we who are not all that wise, that wisdom can't be a matter of merely accumulating knowledge, or comparing it, or tracking it back. We have some feeling that it must be:

practical - it will help us make the most of the situation as it actually is, to do something that can be done;

to do with balance - puting things in their place - so much of this kind of knowledge is useful, so much of this ignorance, so much of this activity, so much of this inactivity, and so on;

must involve a fair amount of self-knowledge - as it's our own biases and blindnesses that skew decisions.

You wouldn't especially expect to find it in an academic, but perhaps it was there in those first "philosophers"

You wouldn't really expect to arrive at wisdom through reading lots of books, or writing essays on them. But the ancient philosophers seem to have been (obviously Socrates is our clearest example) very interested in how to find an education that would develop some wisdom in their young friends. It was by no means a fixed curriculum, and it wasn't necessarily what we would think of as education.

why maths???

2008-05-29T02:30:00.001-07:00

coming down from moni evangelistrias
Originally uploaded by simonsterg.

As a kind of hiking.

Someone once said:

"All the corruption in the world arises from this - someone believed in
someone out of imitation or disbelieved in someone out of imitation."

Well, there should be lots of imitation, , learning a language, learning songs, learning to cook. And then you should be doing things where you don't imitate, where there is a world to explore.

It can be like this with numbers.

An example...

I read the other day that if you square an odd number and take away one you always end up with a multiple of 8.

So, for example you square 5, that's 5 times 5, which is 25. Take away 1 and you get 24 - which is a multiple of 8.

Now doesn't that seem curious? There's an eight hidden away in all those squares of odd numbers? A mysterious eight that nobody put there, but is just waiting for someone to pass by?

This intrigued me. Why was that eight there? Why wasn't it seven, or nine?

So what do you do? You follow the trail...

How many eights are there in the squares?

Make a table:

And then you notice those numbers in the last column...

1 3 6 10 15 21 ...

The triangle numbers!

So, eight triangles are hidden away in the squares.

This must make a good drawing. Hang on...

There they are, the "triangles", eight of them, all fitting neatly into the square!

A hidden terrain opens up!

What next?

Perhaps to prove that however big the square there will always be eight triangles.

And so there is a path you follow, a footpath - only one person at a time. But it could be sociable too if other people were doing that kind of walking.

It's not unique to maths. It could be hunting. Or poetry. Or music. Or science. Or people-watching. Or hiking.

maths

2008-05-08T15:54:00.000-07:00

There's the obvious question: What did they see in maths, as it wasn't practical gain??

I don't know, but I like the few tiny snatches of conversation we hear, famously, from old man Euclid...

§§§§§§§§§ §§§§§§§§§ §§§§§§§§§

New student: "How do I benefit by learning these things?"

Euclid to slave: "Give him threepence, since he has to make a profit out of what he learns".

§§§§§§§§§ §§§§§§§§§ §§§§§§§§§

Ptolomy: "Is there a shorter way to study geometry than your book?"

Euclid: "There is no 'royal road' to geometry."

a figure and a sixpence

2008-05-04T13:37:00.000-07:00

Proclus (450 AD!) says:

Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.

Considering the previous post alongside this, he examined the principles in a material manner too - on a particular beach, with particular stones to play a game, making a circle in a particularly smooth stretch of sand, with particular pupils. He built the road from particular ephemeral arrangements of pebbles all the way to the abstract and immaterial.

Proclus goes on to talk about the motivation for all this:

I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.

on the beach

2008-05-02T07:43:00.001-07:00

beach
Originally uploaded by simonsterg.

returning to Pherecydes, this marvel is reported of him:

"For it is said that he was walking along the sea-shore at Samos, and that seeing a ship sailing by with a fair wind, he said that it would soon sink; and presently it sank before their eyes. "

Is it possibe, that what has come through to us here is Pherecydes talking about the horizon, just a few kilometers away at sea-level...

Could he be talking about how the bottom of a ship disappears, and then the whole ship disappears, when it goes down 'behind' the horizon??

So how about the other 'marvellous story'? -

'At another time he was drinking some water which had been drawn up out of a well, and he foretold that within three days there would be an earthquake; and there was one.'

Wkipedia says that experts are of the opinion that you can't predict earthquakes, but someone suggested to me that perhaps the water in the jug rippled, amplified a tremor that gave Pherecydes good reason to think there could be a bigger one on the way.

Part of what I like about these tantalising tales is that they happen in a place, the first one on a beach. Perhaps Pythagoras was walking on the sea-shore with his teacher? Maybe he reported the story? There's no stopping speculation now... It's this 'outdoors science' that I want to pick up on, to conjure up. With no end of unhistorical speculation of course.

part of a circle

2008-04-26T13:30:00.001-07:00

part of a circle
Originally uploaded by simonsterg.

When we went to the beach one afternoon I took the cord, a ring and a stick. The others watched as I staked the stick in the sand, tied the cord to the ring and put the ring over the stake so it would turn freely.
‘Dorcas, this will be easy for you,’ I said. ‘It’s the hexagon.’
Dorcas stood back from me and announced in a loud voice to the other two:
‘The amazing Leon will now create a perfect hexagon, using only circles and a straight stick.’
‘First of all we need to make the sand flat and get the stones out of the way,’ I said.
We threw the stones towards the sea and smoothed the sand out with sticks.
Then the others stood back while I walked the cord round the stake, digging a circle into the sand just as Pythagoras had done. I made the second and third circle, and then with a stick linked the points to make the hexagon.
Dorcas cheered. She made us all gather up stones and lay white ones along the three circles, and black ones for the hexagon.
After that she made the other two have a go at creating their own hexagons just as I had.
Somehow because she was excited the others agreed straight away, and not only made the hexagons, but were enjoying it as well.
‘This is a kind of game too,’ she said. ‘You have to ask yourself: Can I make a square? Can I make a pentagon? Then the challenge is to make them just using the rope and the stick.’
I’d learnt to make many shapes like this with Anaximenes, but using paper and ink. Over the next weeks the four of us made a lot of them in the sand, and carefully laid stones in the lines of sand to make them last longer. No one moved our stones. The beach became our beach, a giant scroll, its stones not scattered randomly any more but mapping out figures in circles and lines.

tyrant

2008-04-10T07:39:00.000-07:00

It was people like Polycrates that gave us the name 'tyrant'.

The part of Aristotle's Politics which mentions Polycrates is worth quoting:

Tyrannies are preserved two ways most opposite to each other, one of which is when the power is delegated from one to the other, and in this manner many tyrants govern in their states. Report says that Periander founded many of these. There are also many of them to be met with amongst the Persians. What has been already mentioned is as conducive as anything can be to preserve a tyranny; namely, to keep down those who are of an aspiring disposition, to take off those who will not submit, to allow no public meals, no clubs, no education, nothing at all, but to guard against everything that gives rise to high spirits or mutual confidence; nor to suffer the learned meetings of those who are at leisure to hold conversation with each other; and to endeavour by every means possible to keep all the people strangers to each other; for knowledge increases mutual confidence; and to oblige all strangers to appear in public, and to live near the city-gate, that all their actions may be sufficiently seen; for those who are kept like slaves seldom entertain any noble thoughts: in short, to imitate everything which the Persians and barbarians do, for they all contribute to support slavery; and to endeavour to know what every one who is under their power does and says; and for this purpose to employ spies: such were those women whom the Syracusians called potagogides Hiero also used to send out listeners wherever there was any meeting or conversation; for the people dare not speak with freedom for fear of such persons; and if any one does, there is the less chance of its being concealed; and to endeavour that the whole community should mutually accuse and come to blows with each other, friend with friend, the commons with the nobles, and the rich with each other. It is also advantageous for a tyranny that all those who are under it should be oppressed with poverty, that they may not be able to compose a guard; and that, being employed in procuring their daily bread, they may have no leisure to conspire against their tyrants. The Pyramids of Egypt are a proof of this, and the votive edifices of the Cyposelidse, and the temple of Jupiter Olympus, built by the Pisistratidae, and the works of Polycrates at Samos; for all these produced one end, the keeping the people poor. It is necessary also to multiply taxes, as at Syracuse; where Dionysius in the space of five years collected all the private property of his subjects into his own coffers. A tyrant also should endeavour to engage his subjects in a war, that they may have employment and continually depend upon their general. A king is preserved by his friends, but a tyrant is of all persons the man who can place no confidence in friends, as every one has it in his desire and these chiefly in their power to destroy him.

Ikaria 249

2008-03-28T11:18:00.001-07:00

Ikaria 249
Originally uploaded by isl_gr.

This paper says about Ikaria:

"The island was originally densely forested. Icaria was a shipbuilding centre and a charcoal exporting country. Throughout history it has attracted the dominant powers of the eastern Mediterranean, each of which, in its own way, influenced the forest. The flourish cities of antiquity among them Samos, put pressure on its local resources.
These demands have continued through out the vicissitudes of history. Aristotle (4th century BC) mentions in Politics that Polycratis tyrant of Samos, as he strengthened his power and over neighbouring islands, he create a great navy force.
The construction of boats named «samaina» would cause the destruction of Ikaria’s forests."

- although looking through The Politics I can only see one mention of Polycrates, comissioning huge public works to keep his people poor and occupied.

It sounds very likely though. All that wood for the hundreds of boats had to come from somewhere. The paper mentions the trees found here:

"Quercus Ilex (oak tree) is the dominant tree species. Other species include Quercus Pubescens, Arbutus Unedo, Arbutus Andrachne and Phillyrea Latifolia."

+++

This video by Jernej Burkeljca (see here for details) focuses on the present problem with the forest, and attempts to solve it.

dog days

2008-03-27T13:29:00.000-07:00

Sirius the dog star is the brightest star in the sky.

It's usually shining brightly up there, but it has a little break in Summer, when the sun's in it's part of the heavens and its little light is mingled with the sun’s daytime light, making the summer even hotter.

Then as the sun moves away it 'rises'.It becomes visible after its absence in the heat of the summer, the 'dog days'.

In the past, when the stars were the calendar, this was a turning point in the year.Listen to Hesiod in his 'Works and Days'

When the piercing power and sultry heat of the sun abate, and almighty Zeus sends the autumn rains , and men's flesh comes to feel far easier, -- for then the star Sirius passes over the heads of men, who are born to misery, only a little while by day and takes greater share of night, -- then, when it showers its leaves to the ground and stops sprouting, the wood you cut with your axe is least liable to worm. Then remember to hew your timber: it is the season for that work. Cut a mortar three feet wide and a pestle three cubits long, and an axle of seven feet, for it will do very well so; but if you make it eight feet long, you can cut a beetle from it as well. Cut a felloe three spans across for a waggon of ten palms' width. Hew also many bent timbers, and bring home a plough-tree when you have found it, and look out on the mountain or in the field for one of holm-oak; for this is the strongest for oxen to plough with when one of Athena's handmen has fixed in the share-beam and fastened it to the pole with dowels. Get two ploughs ready work on them at home, one all of a piece, and the other jointed. It is far better to do this, for if you should break one of them, you can put the oxen to the other. Poles of laurel or elm are most free from worms, and a share-beam of oak and a plough-tree of holm-oak.

and then again

But when the artichoke flowers, and the chirping grass-hopper sits in a tree and pours down his shrill song continually from under his wings in the season of wearisome heat, then goats are plumpest and wine sweetest; women are most wanton, but men are feeblest, because Sirius parches head and knees and the skin is dry through heat. But at that time let me have a shady rock and wine of Biblis, a clot of curds and milk of drained goats with the flesh of a heifer fed in the woods, that has never calved, and of firstling kids; then also let me drink bright wine, sitting in the shade, when my heart is satisfied with food, and so, turning my head to face the fresh Zephyr, from the everflowing spring which pours down unfouled thrice pour an offering of water, but make a fourth libation of wine.

Too much of Hesiod and he can get a tiny bit self-satisfied, but I love these little snatches of life from back then, the right tree for the right job, the times of year, the plants and sounds... and the stars.