You can think of the square without the center piece. This figure would consist of \(8\) squares, \(4\) being shaded or \(\frac{1}{2}\) shaded. We can think of the center piece of this figure as \(8\) squares again with \(4\) shaded. Each time you count the center piece with \(8\) squares, you have \(\frac{1}{2}\) shaded and \(\frac{1}{2}\) unshaded. This pattern continues forever. Thus the answer is \(\frac{1}{2}.\)
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