Squaring the circle

(UCAS interview brainteaser)

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

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