Sunday, 24 January 2010

How to Paint a Flugel Horn

Can you paint an infinite area with a finite amount of paint?  I am posting this for two reasons - firstly I just love playing with the idea of infinity, and secondly because I want to blog more about infinity in relation to eschatology and faith, and I want to use this as a footnote!  If you really do not like maths please look away now :)

An unwound flugel horn is made by rotating the graph of y=1/x around the x axis from 1 to infinity.

This gives it the unusual property that it has an infinite area, but a finite volume.  Therefore whilst it would require an infinite amount of paint to paint it, which can't be done, you can pour a finite amount of paint into it and tip it out to paint the flugel horn!

Similar logic applies to the problem of the tortoise and the hare.  If the tortoise has a head start then whenever the hare reaches the point at which tortoise was when the hare started then the tortoise has moved further on - so the hare can never overtake the tortoise.

For serious anoraks the maths is below.  All others look away now :)

The area of a flugel horn is the integral of 2π 1/x from 1 to infinity.  This is 2π ln(x) which is infinite.

The volume is the integral of π (1/x)2 from 1 to infinity.  This is -π/x which is finite because the infinity is a divisor.

The thing about the tortoise and the hare is that the infinite number of iterations take place in a finite time and distance.  It is a little like the fact that the sum of 1 + 1/2 + 1/4 + 1/8 .... tends towards 2.

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