Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

Challenge Level

Factorise the equation $$ r^2 - r + {\textstyle{1\over 4}} \sin^2 4\theta = 0$$ and hence plot the graph given by this polar equation.

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NRICH is part of the family of activities in the Millennium Mathematics Project.