In the Zoellner illusion parallel lines are perceived as being tilted in a direction perpendicular to the intersecting oblique segments. The most invoked explanation of this illusion is based on the principle of perceptual enlargement of acute angles. By continuing the oblique segments so that each of them becomes a component of a zigzag path an array of parallel horizontal zigzags is created in between vertical parallel lines. Under these conditions the local geometrical properties of the stimulus are preserved. If the principle of perceptual enlargement of acute angles is valid, Zoellner's tilt should be perceived again. However, differently from what expected, the vertical parallel lines appear clearly undulated. The geometry of the illusion can be simplified by referring to parallel rectangles whose inner surfaces are filled by a zigzagged texture. Phenomenology of the undulation illusion: (i) the illusory weaves are perceived at a global viewing and have a low spatial frequency compared to the much higher spatial frequency of the zigzagged pattern; (ii) as Zoellner's tilt, the undulation illusion is enhanced by rotating the stimulus at 45 deg; (iii) the zigzags do not need to intersect the rectangles but induce undulation at long distance; (iv) the undulations of two contiguous parallel lines appear not parallel but opposite curved with concave and convex alternations; (v) By increasing the width of the rectangles the strength of the undulation increases accordingly. When a black bar is inserted in between each couple of adjacent rectangles another effect emerges: the straight bars appear twisted. If two parallel stripes are now inserted, each of them appears twisted and both intertwine. The roles of geometrical properties of the zigzags and of the parallel lines were studied in five experiments. The results suggest an explanation in terms of global visual mechanisms where eye movements play a role.