From Puzzles to Programming

1. The problem of visual space distortion:

The following figures illustrate how our vision percepts some shapes in a distorted way. Sometimes they are called "geometrical illusions". Can you defy these illusions and visualize the figures in non-distorted way?

Fig. 1. Muller-Lyer illusion

 

 

 

Fig. 2. Hering illusion

 

 

 

 

Fig. 3. Necker cube / Humphrey illusion

 

 

 

 

Fig. 4. A. Distortion of the circles shape

 

 

 

 

 

Fig. 4. B. Bowing of the thick lines inward and towards the center of the concentric squares. Note that the bowing occurs as the lines cross the corners of the squares.

 

 

 

Fig.5. The same diagram as in figure 4. A., but drawn in a much smaller scale.

 

 

 

 

 

Fig. 6. circle-in-square illusion: see the inward bowing of the squares sides.

 

Analysis:

All the figures above are seen in distorted space. But why do we see them distorted. One theory is that the figures suggest depth by perspective, and that this "suggestion" in some way distorts the visual space. This is somehow a generalization but holds true of almost all the known illusion figures.

It is not easy to visualize the above figures in non-distorted way. In figure 1, or Muller-Lyer figure, one line seems shorter than the other while, in fact, both have the same length. In figure 2, Hering illusion, the vertical lines seem to bow inwards and outwards though they are in reality parallel and vertical. In figure 3, the famous Necker cube reverses in depth so that sometimes one face, sometimes another, appears the nearer. This is paradoxical because it looks in depth and yet it is flat on paper. The oblique line is seen slightly bent regardless of which way the cube appears to lie in depth. This is called Humphrey's illusion. In figure 4, the 2 circles appear markedly distorted because the concentric rectangles were viewed unconsciously as indicating depth. In figure 5, the drawing in figure 4 was resized into a much smaller scale so that the spaces among the rectangles are not well seen thus eliminating the depth view and the 2 circles appear much less distorted. In figure 6, the edges of the squares seem to bow inward also because of the concentric circles indicating depth and distance.

The illusion figures may be thought of as flat projections of typical views of objects lying in three-dimensional space. In Muller-Lyer figure, the outward arrows is a typical projection of the corner of a room where the fins represent the intersection of the walls with the ceiling and the floor. The inward arrow represent the projections of the outside corner of a house or a box, the converging lines receding into the distance. Thus, the parts of the figures corresponding to distant objects are expanded and the parts corresponding to nearer objects are reduced in size.

The perceptual phenomena modifies the retinal image. Therefore, to have a non-distorted image is to reduce the scale of the objects in the figure that indicate depth and distance as we did in figure 5. Also, we can make the fins of the arrows of figure 1 shorter as in figure 7:

  

In this figure, both lines are visualized with the same length because the scale of the fins that mark distance is reduced.

Reduction of distortion or illusion is also found among primitive races living in houses without corners. Also, there is the case of a blind man since the first few months of life, but gain sight after operation fifty years later. This man did not have the illusion or distortion of vision after the corneal graft operation suggesting a possible learning mechanism that we get with time from our environment and this give the perspective distortion.

 

 

2. More visual illusions

In the figure below, try not to see the dark-gray dots appearing on the intersections of the white lines between the black squares.