For \(t>0\) and \(n=1,2,...\), let \(L_ n(t)\) denote the length of the lemniscate \(\{\) \(z: | z^ n-1| =t\}\). The lemniscate consists of n components if \(01\). It is natural to ask whether \(L_ n(t)\) is an increasing function of t for each n. Since \(L_ 1(t)=2\pi t\), the answer is affirmative for \(n=1\). \textit{G. Piranian} [Am. Math. Mon. 87, 550-556 (1980; Zbl 0468.30005)] used elementary estimates to give negative answers for the cases \(n=4,5,...\), and invited consideration of the cases \(n=2\) and 3, as well as of other open questions. The present paper addresses and settles several of these questions. 1. \(L_ n(1)\) is given explicitly by a formula involving the gamma function, and is asymptotically equal to 2n. 2. \(L_ 2(t)\) can be evaluated by use of an elliptic integral. 3. For \(n=2,3,...\), \(L_ n(t)\) is increasing for \(0