Triangular tessellations and Fibonacci parastichous spirals were assembled via thermal stress engineering of Ag-core/SiO2-shell microstructures, for which the geometry and topology of the primary core/shells play a pivotal role. Patterns on slightly stressed spherical surfaces are simply triangular tessellations with 12 pentamers, whereas additional pentamer–heptamer chains emerge when more vertices are available (>360) on heavily stressed surfaces, all showing an excellent agreement with numerical solutions to the Thomson's problem. Defect-free triangular lattice can be produced on a pierced spherical cap prepared via the draining effect. Remarkably, Fibonacci spirals of definite chirality, 3 by 5 through 13 by 21, and in both sinister and dexter forms, emerge on the conical supports. These results provide strong confirmation of the mechanical principle for phyllotaxis that the phyllotatic patterns are the least-elastic-energy configuration in a confining receptacle, and meanwhile demonstrate an effective path for the en masse fabrication of patterned structures on curved surfaces.