golden rectangle

A possible insight into the minds of the early mathematics of Pythagoras.

The famous "golden rectangle": here it is made by starting with a small square (1 unit long) and adding another the same next to it. Then alongside those two another, 2 units long, and then another 3 units long.

1, 1, 2, 3, 5, 8, 13 - - - the Fibonacci sequence.

The resulting rectangle approximates to the Golden Rectangle, it's sides the golden ratio.

Here's wikipedia:

"Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of this concept to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

Euclid's Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less." Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio. Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio. Some of these propositions show that the golden ratio is an irrational number."

Whether or not, as some people conjecture, the Parthenon was built with these proportions is hard to say.

Hyperbolic Crochet Coral Reef

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...

♓

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

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

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

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:

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"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.”

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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?

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ancient computer

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

saying of the Pythagoreans:

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

irrational

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

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!