Because the flows in agitated vessels are usually three-dimensional, unsteady, and asymmetric, the explicit interpretation of the fluid mixing mechanism as a deterministic process has been difficult even if the flows are laminar. In this study, we have investigated the shape and dynamical behavior of the streakline from the edge of a paddle impeller, and have proposed that the streakline constructs the template for mixing pattern formation. By examining the template, we have explained the mechanism by which the fluid mixing patterns on r-z, theta-z, and theta-r spaces are formed according to deterministic rules. Streaklines lie on a multi-spiral surface that densely covers the whole mixing region, and each forms many lobe structures. Because the streakline lobes invade into other lobes to make a nesting structure, they can densely cover the whole 3-D mixing region and systematically the fine mixing pattern according to the mixing template constituted by the streakline. Moreover, this template is invariant as long as observations are made at the interval of passage of the paddle impeller.