There do not seem to be any parallel lines here. Or, are there?
Answer
Yes, there are. In each of the three pictures the long lines really are parallel — they only look bent, sloping or curved.
In the first picture the straight vertical lines look bowed because a bundle of rays crosses them.
In the second picture (the café-wall pattern) the long horizontal rows look tilted, but they are exactly parallel.
In the third picture the horizontal lines look curved outwards near the centre.
Check it yourself: Lay the edge of your scale along any one of the lines. It matches the line all the way across — proof that the line is straight. Then measure the gap between two lines at both ends: the two readings are equal, so the lines are parallel.
Q2.
What causes these illusions?
Answer
The background pattern tricks the eye.
Crossing lines at sharp angles. Where a slanting ray meets a straight line, the brain slightly exaggerates the small angle between them. Each crossing pulls the line a little, and the line seems to bend.
Alternating light and dark blocks. In the café-wall picture, black and white tiles are shifted a little in each row. The eye joins the corners of neighbouring tiles and reads a slope that is not there.
A strong centre of radiating lines. The rays give a false sense of depth, and the brain "corrects" the straight lines as if they were far away, making them look curved.
Why it happens: Our eyes do not measure — they interpret. The brain uses the surrounding pattern to guess at depth and direction, and a cleverly designed pattern makes it guess wrong. This is exactly why geometry insists on reasoning and instruments rather than on appearance.
Did you know? That is also the lesson of the whole chapter: two lines may look parallel and not be (Fig. 5.13), or look non-parallel and be parallel. Only equal corresponding angles settle the matter.