Pythagoras

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). 

The two squares still use the same pieces as the large square we started with, so the total area remains exactly the same as it was before we moved things around. 

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. 

Wilson, Keppel and Betty

I call them Wilson, Keppel and Betty. They live inside my brain. I think they are three, but there may be more than one Betty. They are not the Wilson, Keppel and Betty some may remember, if anyone does, although, just the same, they sprinkle sand and scrape it around with their feet.

Betty, however many there are, is not too bad. She is not there all the time. She tries to make you forget things. Like when you know the name of the author of ‘Goodbye to Berlin’, but some cocky little sod from Edinburgh or Oxford shouts out Christopher Isherwood on ‘University Challenge’ while you are still thinking W. H. Auden, which you know is near but not quite right.

I can just about cope with Keppel. He makes your mouth slack and flobby, and blurs your words, but only when you are low on blood sugar. Others say they have not noticed, but that is how it feels to me.

No, Wilson is the worst. He used to put swirling patterns in my eyes. Dr. Hatfield tried to zap him away, but he came back. Mr. Thomson said he would cut him out, but he would not be able to cut all of him out, he would have to leave bits behind.

So Wilson is still there. He now blanks out a space just to the right of my point of focus, and if you can’t see the next             along a line of              then you can only read one word at a             rather than fluently. I should learn mirror-reading, right to left. He also moves words along, and up from the line below, and puts them where you are reading now, slows which letters slows things down even more. And, sometimes, he makes you look at letters for ages before you see what they are, and makes you write an M for a B, or a D for a P, or an S for C. He is a                        total                      mactarp. I have to get the computer to read things out, or Mrs. D.

They have stopped their sand dance for now. So long as I keep taking the Tepmetko Tepotinib they will be quiet. They don’t like it. It makes them ill. It makes me ill too, but not as ill as it makes them. Dr. Brown says that one day they will decide they have had enough and do away with me. It might be this year, but we thought that this time last year, so who knows? Perhaps they realise that if they do away with me, they do away with themselves as well. Mactarps!

https://youtu.be/pkhJpr2zR8s