Thursday, 9 December 2010

A Mathematical Curiosity

I stumbled upon yet another mathematical curiosity on the web. This was something discovered by the Mathematician Hermann Minkowski. If you recall my blogpost about Space and Time you'll realize that this is the same person who reformulated Einstein's Theory of Special Relativity in 4 dimensional spacetime.

Consider the following situation:

A person wishes to travel from the first black spot at the bottom left corner of the diagram to that on the top right. He can only do it by following the grey borders. Assume that one of the square compartments have a length of one unit. If the person travels 6 units east and 6 units north he travels a total of 12 units.

If the person travels along the blue line he again travels 12 units. The figure below is a bit stretched. But with a little bit of imagination you can see that the distance travelled by the person remains 12 units even if the height of on step is reduced so much that it looks like a sawtooth. You can take my word for it but feel free to draw it out and confirm this.

Now Minkowski asked a very interesting question. What if the height of each stair became closer and closer to zero? The path taken by the person become closer and closer to the straight green line. So even when the height of the stair is only 10^-8 units the distance of the path is 12 units. Now's the time a keen mathematician's spider sense starts tingling. What is the length of the green line? According to the Pythagoras theorem it is (6^2 + 6^2)^0.5 = (72)^0.5 which is the same as 6(2)^0.5. But according to the reasoning we just followed it's 12!! So what happened to all that extra length that disappeared when we used the Pythagorean Theorem?

I'll try to give a solution in the next blogpost.

  
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