Squaring the circle

(UCAS interview brainteaser)

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Continued disruption alert! Due to planned major downgrades to pensions of academic staff, members of the University and College Union have been striking at universities around the country. While the initial wave of strike action is over, the dispute has not been resolved, and there is likely to be more disruption after Easter, comparable to the first wave of strikes, unless a solution is found.

You can read more about the reasons for the dispute here, and you can also read an article I was asked to write for the Times Higher Education Supplement here.

It is well known that x^2+y^2=1 is the equation of a circle with radius 1, centred at the origin. Now, suppose we let n be some positive integer. What does the graph of x^(2n)+y^(2n)=1 look like? Try thinking about how x^4+y^4=1 compares to x^2+y^2=1, before thinking about what will happen as n gets larger and larger.

Solution The applet below shows what happens as n increases; drag the slider to the right to see!

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

The shape becomes squarer and squarer as n increases.

Why did we not just look at x^n+y^n=1, do you think?

(The illustration above was made with the mathematics package Geogebra which is free to download and fun to use. Try it out for yourself!)

Contact Details

Dr Sam Marsh
Room G9, Hicks Building
Telephone: 0114 2223792 (internal extension: 23792)
email: s.j.marsh@shef.ac.uk