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arXiv · 2608.22396

On ideal lemniscates, butterflies, and circular waves

Shinya Okabe·Glen Wheeler
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Abstract

We prove the existence of infinitely many lemniscate-like, butterfly-like, and circular-wave critical points for the length-penalised ideal energy. For the lemniscate and butterfly family, we use a staged direct minimisation process to establish existence. A boundary-layer analysis reveals critical points near two Fresnel phases: $3π/4$ modulo $2π$, corresponding to lemniscates, and $7π/4$ modulo $2π$, which are the butterflies. The family of circular waves bifurcates, in a sense, from multiply-covered circles. We use an adapted shooting method to establish their existence.

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Shinya Okabe, Glen Wheeler. 2026-08-23. On ideal lemniscates, butterflies, and circular waves. https://arxiv.org/abs/2608.22396

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