This chapter presents the results of the three experimental studies described in Chapter 3, along with quantitative analyses relating perceptual performance to the mathematical properties of wallpaper groups.

The central finding is that perceptual sensitivity to wallpaper group symmetry is systematically related to — but not simply predicted by — mathematical complexity. Discrimination accuracy in Study 1 increased with the mathematical distance between groups, but with notable exceptions: certain pairs of groups that are mathematically distinct proved perceptually confusable, while other mathematically similar groups were easily distinguished.

The multidimensional scaling analysis from Study 2 revealed a perceptual space with three primary dimensions. The first dimension corresponded closely to the presence or absence of reflective symmetry. The second captured rotational order. The third, unexpectedly, appeared to encode the regularity of the fundamental domain's shape rather than any specific symmetry operation.

Study 3's free-sorting results converged with the MDS analysis. Participants consistently sorted patterns into 4-6 groups, with the primary split based on reflection, followed by rotation. Notably, the perceptual clustering preserved some but not all of the mathematical subgroup relationships, suggesting that the visual system exploits algebraic structure but does not fully recapitulate it.

Together, these results support a model in which symmetry perception operates on a simplified representation of the mathematical structure — one that prioritizes certain symmetry operations (especially reflection) over others, consistent with known properties of early visual processing.