$y^2 = |\cos(\pi*x/2)|$ generates an infinite number of adjacent circles on the line $y = 0$.

Question

http://www.wolframalpha.com/input/?i=y%5E2+%3D+%7Ccos%28pi*x%2F2%29%7C

The generation for the infinite string of circles on $y = 0$.

Is there a relation that generates an infinite number of square adjacent packed circles on the cartesian plane?

How about hexagonally adjacent packing?

Answer 1

There isn't one single relation that will create square or hexagonal packings. However, if you allow piecewise functions, you can easily make "functions".

I believe that the function given above does not give circles, as the Julián Aguirre mentioned. This is because the slope of the above function never approaches infinity (it peaks at $\frac{\pi}{2}$), whereas a function that would output circles would be vertical.