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  1. Euclidean geometry is the fundamental mathematical framework of classical crystallography. Traditionally, layered materials are grown on flat substrates; growing Euclidean crystals on non-Euclidean surfaces has rarely been studied. We present a general model describing the growth of layered materials with screw-dislocation spirals on non-Euclidean surfaces and show that it leads to continuously twisted multilayer superstructures. This model is experimentally demonstrated by growing supertwisted spirals of tungsten disulfide (WS2) and tungsten diselenide (WSe2) draped over nanoparticles near the centers of spirals. Microscopic structural analysis shows that the crystal lattice twist is consistent with the geometric twist of the layers, leading to moiré superlattices between the atomic layers.