Auxetic structures exhibiting negative Poisson's ratios are disclosed including planar auxetic structures formed from x-axis and y-axis strands of selected waveforms. Portions of the auxetic structure referred to as unit cells comprise four joints formed by two adjacent x-axis strands overlapping two adjacent y-axis strands display symmetry about x and y axes. Each unit cell comprises four sub-unit cells formed from segments of ½ wavelength of the wave form. The planar auxetic structures may be formed into three-dimensional auxetic structures or cylindrical or spherical auxetic structures.