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Chiral systems are a class of structures, which may exhibit the anomalous property of a negative Poisson's ratio. Proposed by Wojciechowski and implemented later by Lakes, these structures have aroused interest due to their remarkable mechanical properties and numerous potential applications. In view of this, this paper investigates the on-axis mechanical properties of the general forms of the flexing anti-tetrachiral system through analytical and finite element models. The results suggest that these are highly dependent on the geometry (the ratio of ligament lengths, thicknesses, and radius of nodes) and material properties of the constituent materials. We also show that the rigidity of an anti-tetrachiral system can be changed without altering the Poisson's ratio. The anti-tetrachiral system, with the unit cell shown in red. [ABSTRACT FROM AUTHOR]