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Using linear and weakly nonlinear analysis, and evidence by simulations, we demonstrate how the domain shape can fundamentally change the expected bifurcation structure. Specifically, on the square domain we are able to derive pitchfork bifurcations for stripe and spot solutions, as well as show that both branches cannot bifurcate to produce stable patterns. This compares with the case of the equilateral triangle domain that causes the Turing bifurcation to be generically transcritical and, in some cases, none of the bifurcating branches are stable. Moreover, we find a monotonically increasing, but nonlinear relationship, between the minimal bifurcation area and the number of edges. Thus, patterns can occur on triangles with much smaller areas than circles. Overall, this work raises questions for researchers who are simulating applications on domains with simple shapes. 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