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Elcric 'selaht

Enter a geometrical theorem - by the backdoor!
by Klaus Kohl

I. To the amateur - the enthusiast - the student:
"Get yourself a square board, a hammer and two nails! Drive the two nails... No, not into tht board! Drive them a bit wide into the ground, into the wall, another board, not too near to each other, but so near that the square board will never pass through the gap!"
That's the first step. And now act as if you would try to pass that board through the gap. Push, turn it left and right but be shure that it is always rubbing the nails. Push it, turn it to and fro and look at its moving. How does it move? Is there a point in rest? A fixed turning point? Its center perhaps? No. But how does that center move? How do the other points of the square? It is not easy to get these lines. Yet it is easy by making a hole into the square and stick a pencil (better a colored marker) through it. That will give nice patterns, bows and so on. Nearly impossible to guess the form before. Calculable? Of course it should be possible. But difficult...
Now look at the edge of the square! What is its way? It looks like walking a real circle bow. Really? Exactly? Take your compasses and compare - indeed! And the diameter of this semicircle is the distance of the two nails. You can pour sand on the floor and the square will clear away a clean semicircle.
Is the square a must? Try a rhomb. Cut it from cardboard, that is easy to do. It will form by its edge a circle, too. But more than a semicircle with its sharp edge and less with the other. The diameter? Difficult...
Is the square a must to have a rectangle? Not at all, as all behind the nails does not matter. Any rectangle will do as well as a set square.
Rectangular and circle... wasn't there something about Thales' circle?

II. To the professional who has to do it (eagerly, I hope):
Yet another way to get a circle, without compasses or a cord. No center point with a defined distance.
It's a rather unusual way to enter the theme. Surprising to all who are no experts to see the result. You may imagine old (better young) Thales at the beach trying to push a square board through the gap between two poles and racking his brain over the circle formed in the sand. At this point it does not matter at all that we don't get closer to the historical Thales. (But - I am sure he was not born as a dignified bald-headed bearded old man - he was young and active when he got his ideas!) He will ever be envelopped in the pre-socratical fog as well as the answer to the question what he really might have revealed about 'his' circle [1].
Martin Wagenschein estimated such surprising entrances into a theme. Some of them have been developed to real pieces of teaching art [2]. Thales' circle was named by Wagenschein "the Thales Phenomenon" [3]. His imagination: You are a spectator sitting in the semicircle of a greek theater. Looking from one edge point to the other you will always make a turn of rectangular shape. Wagenschein's proof works without a circle line at all. The problem is reduced to two even-sided triangles sharing equally the diameter of the circle and thus having a common side as long as its radius. "By Zeus!" Menon's slave would have cried out in Platon's text, when he had observed the conclusion: There must be a right angle. (Doubling the square takes less assumptions, Sokrates could go faster). This here is no teaching art at all, at best an overture. Now should come the proven workout, leading to a final, perhaps the peripheral angles always being half the shape of the central angle by the cord.

III. To the curious explorer
it is really not necessary to carry owls to Athens. The fruitbearing branches aside are long discovered! Twice the word "difficult" showed up in the first part. Much fun by harvesting!


Published in: Mathematik in der Schule, Heft 7/8 1998 p. 385 ff.
Pädagogischer Zeitschriftenverlag, Berlin