Polymersomes are vesicles formed by the self-assembly of amphiphilic copolymers in water. They represent one of the most promising alternatives of natural vesicles as they add new possibilities in the amphiphiles' molecular engineering of aqueous compartments. Here we report the design of polymersomes using a bottom-up approach wherein self-assembly of amphiphilic copolymers poly(2-(methacryloyloxy) ethyl phosphorylcholine)-poly(2-(diisopropylamino) ethyl methacrylate) (PMPC-PDPA) into membranes is tuned using pH and temperature. We report evolution from disk micelles, to vesicles, to high-genus vesicles (vesicles with many holes), where each passage is controlled by pH switch or temperature. We show that the process can be rationalized, adapting membrane physics theories to disclose scaling principles that allow the estimation of minimal radius of vesiculation as well as chain entanglement and coupling. This approach allows us to generate nanoscale vesicles with genus from 0 to 70, which have been very elusive and difficult to control so far.