Guide Hidden Harmonies: The Lives and Times of the Pythagorean Theorem

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A squared plus b squared equals c squared. It sounds simple, doesn't it? Yet this familiar expression is a gateway into the riotous garden of mathematics, and sends us on a journey of exploration in the company of two inspired guides, acclaimed authors Robert and Ellen Kaplan.

With wit, verve, and clarity, they trace the life of the Pythagorean theorem, from ancient Babylon to the present, visiting along the way Leonardo da Vinci, Albert Einstein, President James Garfield, and the Freemasons-not to mention the elusive Pythagoras himself, who almost certainly did not make the statement that bears his name.

How can a theorem have more than one proof? Why does this one have more than two hundred-or is it four thousand? The Pythagorean theorem has even more applications than proofs: Ancient Egyptians used it for surveying property lines, and today astronomers call on it to measure the distance between stars. Its generalizations are stunning-the theorem works even with shapes on the sides that aren't squares, and not just in two dimensions, but any number you like, up to infinity.

And perhaps its most intriguing feature of all, this tidy expression opened the door to the world of irrational numbers, an untidy discovery that deeply troubled Pythagoras's disciples. Physical description; xii, p. Notes; Includes bibliographical references p.

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And there on page 17 was Miss Coolidge's proof. Across from the Notes and Queries. What is the guinea n sect. Try googling it it's not there for proof however was not as it appears in Loomis who had clearly exercised his editorial powers over it. He was somewhat high in manner. We give her original proof in the book with all its extraneous steps. What's particularly fascinating about this proof is that Miss Coolidge begins it geometrically making and comparing shapes.

Then turns to calculating with letters when perhaps she can't quite figure out what's left and what's wanting. You might think that being blind Miss Coolidge would have resorted to abstraction as early as possible. But in fact she delays it as long as she can trying to stay true to the spirit of geometry. Having followed for proof if we now follow her we may better understand the play between obstruction and different sorts of sensory information.

But how find the woman behind Lewis's brief Miss Coolidge. It struck us that Coolidge is a good post a. Might she not then have been a student at the Perkins School for the blind. We e-mailed Jan Seymour for their research librarian who answered. That was inspired guesswork. Emma a Coolidge was a Perkins student.

Robert Kaplan and Ellen Kaplan - Events - Harvard Book Store

She was born on August 4th in Sturbridge mass and she lost her sight from whooping cough when she was a year old. She could only. And none of the remedies her parents tried having her wear gold earrings putting talcum powder into her eyes blistering her temple with poison flies helped. After graduating from after graduating from Perkins she studied at Wellesley for a year then returned as a teacher to Perkins where one of her students is said to have been Annie Sullivan.

Later Helen Keller's tutor Emma married had a daughter wrote children's books taught music in her New Hampshire village school sewed knitted could kill and dress a fowl for dinner. If her husband was away and boldly went out alone tying white rags to doorways so she could find her way from place to place. Isn't this what we see in her proof. Or what you'll see in her proof if you look in the book where the proof is catching the cure a skewer of prominent shapes but navigating otherwise by those abstract relations with which practice and memory furnish the mind.

What sort of imagination this involves may matter here. The blind mathematician Louis Antoine was led by his advisor. Oh you live a study two and three dimensional typology is quoting the bag in such a study. The eyes of the Spirit and the habit of concentration will replace the lost vision. His equally outstanding compatriot Bernard Morva has been completely blind since age 6 was asked how he knew the correct sign in a long and difficult computation.

More tellingly more un distinguishes between what he calls time like and space like mathematical imagination and surprisingly says that he excels at the latter.


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A problem with picturing geometrical objects is that we tend to see only their outsides which hide what might be complicated within MOA who works with extremely intricate objects in three dimensions has taught himself how to pass from outside to inside from one room to another. Our spatial imagination he says is framed by manipulating objects.

You act on objects with your hands not with your eyes. So being outside or inside is something that's really connected with your actions on. Think of Emma sewing and knitting or killing and dressing fellow. Might Emma's extraneous calculations have come from an intrusion of the timelike into her sounder space like imagination and is the tactile yet one more intermediate between being and becoming.

Spatial imagination.

All possible pythagorean triples, visualized

What does this mean. What does it mean when physicists talk of string theory and see these strings in leavened dimensional space 11 or 9. Which is it. What does it mean when that magicians toss around 12 and dimensional space in their coffee fueled conversations. They aren't superhuman like us. They can only see three dimensions. So let's go back for centuries. This is from the book in 17th century the master regular Yohannes fellow Harper discovered a spatial generalization of the Pythagorean theorem and saw how it will lead him to a deeper understanding of six hundred sixty six which he already knew was divine rather than diabolic while the margins of our minds are too narrow to contain fall Harbor's cabalistic proof we will unveil to you his rectilinear pyramid.

Analogy said his contemporary Yohannes Kepler is the greatest of my masters moving by analogy from squares on the lengths of a right triangle sides just squares on the areas of a right pyramids basis.

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He came up with this beautiful generalisation given such a pyramid three of its edges meeting. The sum of the squared areas of these faces a B and C equals the square of the area on the last hypothenuse space d.. That is a squared plus B squared le c squared equals D squared. If not quite up there with still pretty wonderful. In the winter of 16 Take heart. Passing through them. They will have met fell harder. A local legend has it that one night dazzled by Jake Hart's brilliance fall hover reached out. I picture them as sitting opposite sides of his fireplace in a stuffy comfortable room with weather freezing outside the harbor reached out and touched his guest to make sure that he was human and not an angel.

Wouldn't you have done this much where your visitor to have mentioned that is there are also held in four dimensions. What do you make of that what do you make of this talk of dimensions beyond those we can see. Is this meaningful is it meaningless is it a game is it the most important thing in the world.

Any Of The Above all or none or anything you'd like to talk about. Whether it's the Pythagorean theorem and its consequences it's applications. The still unsolved open problems in it is that a question Rick. Just looks like you're right.

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It looks like so what. What is a proof. What constitutes a proof. Why are not the algebraic manipulations with letters and symbols an equal size Why are those.


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  • Why are all proofs illusions. What isn't an illusion. Yes yes. We do with 8 year olds. Me read. But I so I see your point here is. You write your name right almost as an origami or using of origami like proofs for Angle trisection or supposedly of legitimate out of.

    Robert Kaplan and Ellen Kaplan

    Oh yes oh yes yes yes yes yes yes yes yes. Where oh oh oh oh oh. Not one that means very well. But remember there are there are cultural reasons behind at the time. This is the third century A. Three was the circumference of a circle of Diana radius 1 and 4 was the circumference of the square of side once a three and four key numbers and thought of well as my favorite number.

    So if it's true of that key mystical triangle Well of course is going to be true of other triangles. And there was another factor in Chinese in Chinese thinking about this you'll see this in the book that for them the only triangle the head area was a right triangle because you have height times length of base give you area other triangles more or less head area.

    Rather like the problem with this 64 by 65 rectangle. But I think the you've touched on a real nerd in mathematics. How does one validate a proof. Chris custodian custodian steps us who takes who who's in charge of the. What finally validates anything in the century that's passed and the one that we're in the midst of the foundational issues or devastating juggles seems to have proven that with nothing matics as we know it there will be true statements that you cannot prove and yet they're in some sense true.

    What does that mean. And when you take a proof which has all the paraphernalia of rigor attached to it with the questions and symbols reduced to definitions and axioms well why take these actions rather than those a big issue with a bit like me and theorem where a service is other than a plain analogues of the theorem holds but not a square of the speed squared equals C squared. Perhaps something with CO signs will hold instead on a sphere or on a and inverse fear looking like looking like.

    Yeah on that you get something with hyperbolic co signs which is the equivalent of the Pythagorean theorem. So which surface. Does the theorem hold. What is the mean. I mean we don't actually. Therefore the shorthand way of describing what's going on is with what's marvelous about anything like this is that we can hear a stance that Leader Reid would actually hinder to find out things from you. So we would love to hear more of your ideas about what constitutes proof in mathematics.

    It allows them to get the basic idea when you start something you know you know to be true. Oh this is this is. Yes right. So I don't know Station of Lenny. Its generalizations are stunning--the theorem works even with shapes on the sides that arent squares, and not just in two dimensions, but any number you like. And perhaps its most intriguing feature of all, this tidy equation opened the door to the world of irrational numbers, an untidy discovery that deeply troubled Pythagoras' disciples.

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