Fig. 3: Real fish exhibit vortex phase matching (VPM).

a Example frames from a video recorded in the flow tank, overlaid with the result of deep-learning-based body posture tracking. The phase of the undulations of each fish was determined by the motion of the tail tip (measured at the caudal peduncle). b–e Occurrences of observed swimming relationships (probability density function) as a function of phase difference Φ and front-back distance D between leader and follower illustrated by colour-coding for one case with intact vision and lateral line (V+LL+), three cases with impaired vision (V-LL+), the lateral line (V+LL-) or both (V-LL-) (b), and all pooled data (c; V+LL+: n = 6, V+LL-: n = 5, V-LL+: n = 3, V-LL: n = 2; n given is the number of pairs). Data are duplicated twice along the phase difference axis to more clearly demonstrate the periodic pattern. Theoretical predictions (d) are derived by transforming observed swimming characteristics (including frequency f, front-back distance D and flow speed u) using our model (Eq. (2)) with Φ₀ = \({\Phi }_{0}^{* }\) = −0.2π. \({\Phi }_{0}^{* }\) is determined by comparing experimental data and theoretical predictions after Von Mises fitting with a periodic least square fitting algorithm (Methods section). Randomised data (e) are generated by swapping two fish postures respectively from two randomly selected trials within same treatment (Supplementary Note 4).