Chapter 6 - The Influence of Angles:
Spiral Optical Illusion, Twisted Chord Optical Illusion, Angle Optical Illusions and More

Some other optical illusions are now presented to demonstrate further the effect of the presence of angles. Undoubtedly, in some of these optical illusions, there are other causes which contribute more or less to the overall result. In Fig. 47 a series of concentric arcs of circles end in a straight line. The result is that the straight line appears to sag perceptibly. Incidentally, it may be interesting for the reader to ascertain whether or not there is any doubt in his mind as to the arcs appearing to belong to circles. To the author the arcs appear distorted from those of true circles.

Fig. 47. - A straight line appears to sag.

In Fig. 48 the bounding figure is a true circle but it appears to be distorted or dented inward where the angles of the hexagon meet it. Similarly, the sides of the hexagon appear to sag inward where the corners of the rectangle meet them.

Fig. 48. - Distortions of contour due to contact with other contours.

The influences which have been emphasized apparently are responsible for the optical illusions in Figs. 49, 50 and 51. It is interesting to note the disappearance of the optical illusion, as the plane of Fig. 49 is varied from vertical toward the horizontal. That is, it is very apparent when viewed perpendicularly to the plane of the page, the latter being held vertically, but as the page is tilted backward the optical illusion decreases and finally disappears.

Fig. 49. - An illusion of direction.

The optical illusions in Figs. 50 and 51 are commonly termed "twisted cord" effects. A cord may be made by twisting two strands which are white and black (or any dark color) respectively. This may be superposed upon various backgrounds with striking results. In Fig. 50 the straight "cords" appear bent in the middle, owing to a reversal of the "twist." Such a figure may be easily made by using cord and a checkered cloth.

Fig. 50. - "Twisted-cord" optical illusion. These are straight cords.

In Fig. 51 it is difficult to convince the intellect that the "cords" are not arranged in the form of concentric circles, but this becomes evident when one of them is traced out. The influence of the optical illusion is so powerful that it is even difficult to follow one of the circles with the point of a pencil around its entire circumference. The cord appears to form a spiral or a helix seen in perspective.

Fig. 51. - "Twisted-cord" optical illusion. These are concentric circles.

A striking optical illusion is obtained by revolving the spiral shown in Fig. 52 about its center. This may be considered as an effect of angles because the curvature and consequently the angle of the spiral is continually changing. There is a peculiar movement or progression toward the center when revolved in one direction. When the direction of rotation is reversed the movement is toward the exterior of the figure; that is, there is a seeming expansion.

Fig. 52. - A spiral when rotated appears to expand or contract, depending upon direction of rotation.

Angles appear to modify our judgments of the length of lines as well as of their direction. Of course, it must be admitted that some of these optical illusions might be classified under those of "contrast" and others. In fact, it has been stated that classification is difficult but it appears logical to discuss the effect of angles in this chapter apart from the divisions presented in the preceding chapters. This decision was reached because the effect of angles could be seen in many of the optical illusions which would more logically be grouped under the classification presented in the preceding chapters.

In Fig. 53 the three horizontal lines are of equal length but they appear unequal. This must be due primarily to the size of the angles made by the lines at the ends. Within certain limits, the greater the angle the greater is the apparent elongation of the central horizontal portion. This generalization appears to apply even when the angle is less than a right angle, although there appears to be less strength to the optical illusions with these smaller angles than with the larger angles. Other factors which contribute to the extent of the optical illusion are the positions of the figures, the distance between them, and the juxtaposition of certain lines. The optical illusion still exists if the horizontal lines are removed and also if the figures are cut out of paper after joining the lower ends of the short lines in each case.

Fig. 53. - Angles affect the apparent length of lines.

In Fig. 54 the horizontal straight line appears to consist of two lines tilting slightly upward toward the center. This will be seen to be in agreement with the general proposition that the sides of an angle are deviated in the direction of the angle. In this case it should be noted that one of the obtuse angles to be considered is ABC and that the effect of this is to tilt the line BD downward from the center.

Fig. 54. - The horizontal line appears to tilt downward toward the ends.

In Fig. 55 the horizontal line appears to tilt upward toward its extremities or to sag in the middle. The explanation in order to harmonize with the foregoing must be based upon the assumption that our judgments may be influenced by things not present but imagined.

Fig. 55. - The horizontal line appears to sag in the middle.

In this case only one side of each obtuse angle is present, the other side being formed by continuing the horizontal line both ways by means of the imagination. That we do this unconsciously is attested to by many experiences. For example, we often find ourselves imagining a horizontal, a vertical, or a center upon which to base a pending judgment.

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