Decipher the greeting card $( X^2 +Y^2 -1 ) ^3 - X^2 Y^3 = 0$

Using Maple it is:

 [> with(plots):
    implicitplot((x^2+y^2-1)^3-x^2*y^3 = 0, x = -10 .. 10, y = -10 .. 10);

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http://mathworld.wolfram.com/HeartCurve.html

Have a look at the second graph here

It is a cardiod curve.

Your friend had a great equation for his greetings, check the graph below

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