|If more of us valued food and cheer and song above hoarded gold, it would be a merrier world. - J.R.R. Tolkien|
Chapter 6 - The Influence of Angles:
|Fig. 42. - The effect of two angles
in tilting the horizontal lines.
In Fig. 42 the oblique lines are added to both horizontal lines in such a manner that A is tilted downward at the angle and B is tilted upward at the angle (the letters corresponding to similar lines in Fig. 41).
In this manner they appear to be deviated considerably out of their true straight line. If the reader will draw a straight line nearly parallel to D in Fig. 41 and to the right, he will find it helpful. This line should be drawn to appear to be a continuation of C when the page is held so D is approximately horizontal.
This is a simple and effective means of testing the magnitude of the optical illusion, for it is measured by the degree of apparent deviation between D and the line drawn adjacent to it, which the eye will tolerate. Another method of obtaining such a measurement is to begin with only the angle and to draw the apparent continuation of one of its lines with a space intervening. This deviation from the true continuation may then be readily determined.
A more complex case is found in Fig. 43 where the effect of an obtuse angle ACD is to make the continuation of AB apparently fall below FG and the effect of the acute angle is the reverse. However, the net result is that due to the preponderance of the effect of the larger angle over that of the smaller. The line EC adds nothing, for it merely introduces two angles which reinforce those above AB. The line BC may be omitted or covered without appreciably affecting the optical illusion.
|Fig. 43. - The effect of crossed lines upon
their respective apparent directions.
In Fig. 44 two obtuse angles are arranged so that their effects are additive, with the result that the horizontal lines apparently deviate maximally for such a simple case. Thus it is seen that the tendency of the sides of an angle to be apparently deviated toward the direction of the angle may result in an apparent divergence from parallelism as well as in making continuous lines appear discontinuous. The optical illusion in Fig. 44 may be strengthened by adding more lines parallel to the oblique lines. This is demonstrated in Fig. 38 and in other illustrations. In this manner striking optical illusions are built up.
|Fig. 44. - Another step toward the
Zollner optical illusion.
If oblique lines are extended across vertical ones, as in Figs. 45 and 46, the optical illusion is seen to be very striking. In Fig. 45 the oblique line on the right if extended would meet the upper end of the oblique line on the left. However, what appears to be the point of intersection looks a little lower than it really is. In Fig. 46 the oblique line on the left is in the same straight line with the lower oblique line on the right. The line drawn parallel to the latter furnishes an idea of the extent of the optical illusion. This is the well-known Poggendorff optical illusion.
|Fig. 45. - The two diagonals would
meet on the left vertical line.
The upper oblique line on the right actually appears to be approximately the continuation of the upper oblique line on the right. The explanation of this optical illusion on the simple basis of underestimation or overestimation of angles is open to criticism. If Fig. 46 is held so that the intercepted line is horizontal or vertical, the optical illusion disappears or at least is greatly reduced. It is difficult to reconcile this disappearance of the optical illusion for certain positions of the figure with the theory that the optical illusion is due to an incorrect appraisal of the angles.
|Fig. 46. - Poggendorff's optical illusion.
Which oblique line on the right is the
prolongation of the oblique
line on the left?
According to Judd [A Study of Geometrical Illusions, C. H. Judd, Psych. Rev. 1899, 6, p. 241.], those portions of the parallels lying on the obtuse-angle side of the intercepted line will be overestimated when horizontal or vertical distances along the parallel lines are the subjects of attention, as they are in the usual positions of the Poggendorff figure. He holds further that the overestimation of this distance along the parallels (the two vertical lines) and the underestimation of the oblique distance across the interval are sufficient to provide a full explanation of the optical illusion. The disappearance and appearance of the optical illusion, as the position of the figure is varied appears to demonstrate the fact that lines produce optical illusions only when they have a direct influence on the direction in which the attention is turned. That is, when this Poggendorff figure is in such a position that the intercepted line is horizontal, the incorrect estimation of distance along the parallels has no direct bearing on the distance to which the attention is directed. In this case Judd holds that the entire influence of the parallels is absorbed in aiding the intercepted line in carrying the eye across the interval. For a detailed account the reader is referred to the original paper.
Chapter 6 - The Influence of Angles:
Zollner Optical Illusions
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Chapter 6 - The Influence of Angles:|
Spiral Optical Illusion, Twisted Chord Optical Illusion, Angle Optical Illusions and More
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