It was well-known to the ancient Egyptians, that a triangle with sides of 3, 4 and 5 units makes a right-angle. The Babylonians also knew this four thousand years ago, as they did in India. They used it to measure out precise squares and verticals. I would not be surprised if the ancient Tom Stephenson used it too.
Rotate such a triangle four times by ninety degrees, and you are back to where you began. Put four of them together as shown below on the left and it makes a perfect square. The one on the right is the same with the middle bit filled in.
This works for any right-angled triangle, not just those of size 3-4-5.
The sides of the square are equal in length to the long sides of the triangles.
I am now going to move the top two triangles, top right to bottom left, and top left to bottom right. Hopefully, the arrows and numbers help make this clear. It results in an L-shape.
The L-shape uses the same pieces as the large square we started with, so the overall area remains exactly the same.
The L-shape can be split into two squares, a large one and a small one, as below. The sides of the smaller square are of the same length as the shortest side of the original triangle (see right-hand side). The sides of the larger square are of the same length as the third side (see left-hand side).
In other words, the area of the square formed around the longest side of the triangle is the same as the combined areas of the squares formed around the other two sides.
Or as Pythagoras put it in 550 B.C.: “The square on the hypotenuse is equal to the sum of the squares on the other two sides.”
Isn’t that just exquisite. They never showed me that at school.
Pythagoras may have discovered this by moving triangles around in the same way, one of the first to express the structure of nature as numbers, and advance understanding from the world of fact into the world of proof. He offered a hundred oxen to the muses in thanks for the inspiration (Jacob Bronowsti: The Ascent of Man).
And once you know this is true for any right-angled triangle, you can work out how much timber and how many tiles you need for your pitched roof, and are on the way to the kind of trigonometry that allows you to manipulate three-dimensional images on your computer screen.
You lost me at "It was well-known..." and then I found myself recalling my aptly named old Maths teacher - Mr Borman! To me a triangle is something you play with a stick.
ReplyDeleteHow do you manage to find your way out walking with a map and compass?
DeleteEasily.
DeleteDo you have to turn round until the map is the same way round as the terrain?
DeleteI'm glad I'm not the only one!
ReplyDeleteYou must try harder, Janice.
DeleteDear Tasker, I read it all through, and it gripped me as much or as little as our math lessons..:-) (Those were always a welcomed occasion for boys to try to explain it to me, haha)
ReplyDeleteTill the end I hoped I might find a hint about the mysterious sentence: "I would not be surprised if the ancient Tom Stephenson used it too" - but no - or I lost it...
I was not trying to show any cleverness, but rather the opposite. It's one of those things I always accepted because I was told it, but never let on I didn't understand why it was so.
DeleteTom may take up the bait.
When I was in school, algebra and trig were so painful. I just hate them, I didn't understand, my parents were livid that after getting such good grades in elementary school, I was struggling. There was a lot of yelling, and a lot of smacking.
ReplyDeleteYears later, I was in the army, and doing CQ duty. An aide said, "You're smart. I cannot figure out this algrebra." I said that Math was never a good subject for me. I sat down with his book, and much to my surprise, I figured it out immediately, and was able to teach him. It seemed so perfectly logical.
When I took my own college classes, I took a math logic class, and I aced it. I always wonder how much of my math struggles were fear based? But I remember being intrigued by the pythagorean theorem. A squared + B squared = C squared. Years after the fact, when we had to build our first set of stairs, Tim was struggling. I sat down and figured it for him, using Pythagoras' wisdom. I probably did it the hard way, but my calculations and drawings were right.
Yes, I think it's easy to assume we can't do somethings because of early difficlty, especially if others seem to zoom ahead with ease. Until I saw this, I could not imagine that the proof of Pythagoras was somethimg I could do.
DeleteI bet you use this sort of thing a lot in building.
It's all Toblerones and pyramids to me Tasker.
ReplyDeleteBut how much Toblerone can you grow in your polytunnel?
DeleteI have always wondered why so many people 'HATE' maths. is it how it is traditionally taught? It is simply a way to explain the world around us and as you have amply demonstrated here beautiful in its simplicity when presented another way.
ReplyDeleteI found it so elegant I got quite excited. We have beautiful minds.
DeleteI understood the square of the hypotenuse is equal to the sum of the squares on the other two sides at school but I understand less of your diagrams. I remember Tom once saying he had bought a protractor but he may kill me for saying that.
ReplyDeleteI've never had problems with the basic idea, e.g. that 9 + 16 = 25, but I could not have proved why it was so always true without these diagrams. I would not have tried to understand a mathematical proof.
DeleteSchool nightmares trig gered.
ReplyDeleteMaybe, but this almost makes me want to study maths again.
DeleteI learned this in eighth grade from Miss What Was her Name? I loved geometry.
ReplyDeleteA geometrical proof of a mathematical concept.
DeleteYou explained and illustrated that very well. I never took trigonometry but I loved geometry as it was like working puzzles to me.
ReplyDeleteIt makes it so clear to me too, but seemingly not to all. It's like cutting up patterns and rearranging them.
DeleteGeometry and trigonometry were the parts of maths lessons that I really understood, and liked, while I struggled with algebra most of the time. And typical Meike-style, if I didn't understand something right away, I lost interest and a sort of mental boycot took place. This is a shame, because I actually DO have a scientist's mind - I just never really applied myself to sciences when I was young enough to give my life that direction.
ReplyDeleteYou say "They never showed me that at school." That surprises me! We learned all that at school. But of course, my school days were not only in a different country, but also in a different decade.
I think our teacher simply told us a2 + b2 = c2 without explaining why it is so. I used to struggle with some maths concepts too - I remember trying to do calculus - and had the same tendency as you which was to give up because it needed a bit of effort. I would be very different now.
DeleteSadly I am lost but they make pretty patterns. ;)
ReplyDeleteCommenters appear to fall into two distinct camps.
DeleteThe shading was the accidental result of converting .png to .jpg, but I like it too.
It never occurred to me to prove Pythagoras any further than this: The fact that I could draw a right triangle, measure the two sides, do the calculation and discover that my measurement and my calculation of the hypotenuse were in agreement. The squares are a fascinating foray farther into it. I spent a surprising bit of time pondering this.
ReplyDeleteYou get exactly where I am coming from. I've spent more than a surprising amount of time drawing those diagrams. I am now wondering whether it can be done with pi and circles.
DeleteMy brain is not up to following this today, I'm afraid! I'll take your word for it, tho. :)
ReplyDeleteI have days like that too. Wrote this on a good day.
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