Question on a simple geometry subject.

[El barto]

So my friends and I are arguing about a theory I've come up with. Given a square, you can divide it into 4 squares, and it will the same distance from one corner to the other if you go through the big square via the sides of the smaller squares as it would going around the perimeter of the square. Now, if you divide this square into smaller squares (16, for the sake of argument) it will take 8 side lengths to get from one corner of the big square to an opposite corner if you go through the outside of the big square or the inside of the big square. So no matter how many time you divide a square, the distance along the "zig-zagged" line from one corner to the other will be the same as half of the perimeter of the square. Now I said that when a square is divided into an infinite amount of smaller squares, the "zig-zagged" line through the square would become a perfectly straight line, but would still be the same length as two of the larger square's sides. Obviously this is a paradox because in a right triangle the legs have to add up to a greater value than the hypothesis. Can anyone verify if I'm imagining this right? At an infinite number of divisions, will a line form from a corner to the opposing corner, and will it be the same distance as half of the perimeter?

 

[yenchin]

You have discovered your own variant of the limit paradox:
http://mathpages.com/home/kmath063.htm

 

[Joskoplas]

That is one of the coolest things I've seen in a long time. Thanks to both of you!

 

[El barto]

Thanks Yenchin that explained a lot. So, the diagonal zig-zag does eventually become a straight line (when the number of divisions reaches infinity), it just doesn't equal the distance of two sides of the square anymore?

 

[yenchin]

Yes. That is, the limit of a sequence may not have all the properties of the terms in the sequence. For example, one can construct a sequence in which each term is a rational number, but the limit that it approaches is irrational. It is basically the same thing you think of here.

 

[dgeomprof]

The geometry that you are using for the sequence of curves is called the taxi cab geometry. It is true that this curve approaches the diagonal in the point-wise limit. The reason for the apparent paradox is that the taxi cab curve you are using is not related to the arc-length definition and so you should not expect it's length in the limit to be the arc-length of its point-wise limit.

Another interesting system curves are the 'space filling curves' (see wikipedia for examples) The limit set of a space filling curve can be the entire square which of course is not measured in length, but in area.