We obtain new BPS solutions of six-dimensional, N = 1 supergravity coupled to a tensor multiplet. These solutions are sourced by multiple "superthreads" carrying D1-D5-P charges and two magnetic dipole charges. These new solutions are sourced by multiple threads with independent and arbitrary shapes and include new shape-shape interaction terms. Because the individual superthreads can be given independent profiles, the new solutions can be smeared together into continuous "supersheets", described by arbitrary functions of two variables. The supersheet solutions have singularities like those of the three-charge, two dipole-charge generalized supertube in five dimensions and we show how such five-dimensional solutions emerge from a very simple choice of profiles. The new solutions obtained here also represent an important step in finding superstrata, which are expected to play a role in the description of black-hole microstates, due to their ability to store a large amount of entropy in their two-dimensional profile.
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