On the Ponzo effect
By Rustan Andersson
When looking at the compositions of lines that we know as the Ponzo effect we experiences two readings that are fundamental different; the lines on the ground in the called the object plane or if you are looking at a picture with projecting lines in 2D projection plane. The second case can be read as a representation of the former.
Describe a plausible explanation.
The immediate and obvious explanation of the Ponzo effect when read as a picture is the resemblance with the linear perspective, a renaissance invention and a representation of the world in a logical coherent 2D plane. I think one can argue that this geometrical construction as an invention is a result of how we experience of vision and in a way circumvent the inverse problem. The geometrical foundation is build on Euclid’s Optics in Elements and founded on the Platonic idea that vision is caused by discrete light rays that emanates from the eye. From this notion Euclid postulated that the visual rays created a visual cone with the apex in the eye and was the first mathematical theory on vision. From this theory he deduced that the angle of the cone determined how big the object was in front of the spectator; lager angles implied closer and bigger object. Renaissance artists like Brunelleschi, Alberti, Leonardo and later Dürer used Euclid in their work on perspective but changed the construction fundamentally by creating a picture plane and vanishing point on the line of infinity where al parallel lines will eventually meet. This way of constructing space has dominated western art for nearly 400 years.
So when we see two lines that in the picture plan tend to converge we immediately in cultural sense “know” that these two lines ideally will meet in “vanishing point at the line of infinity“.
This situation embrace two fundamental postulates in two different geometries; in the Euclidian geometry the fifth postulate says that to parallel lines never meet and in the projection geometry that parallel lines always will meet in a point. One can easily claim that these postulates could be important to explain the Ponzo effect, and the so-called perspective hypothesis.
But the paradox is that these “projections” on the retina are incompatible and neither of them will tell us anything about the reality.
We will empirical never be able to prove that the parallel lines are parallel in either the projective plane or in the reality.
Outline an experiment that you could perform to test you
Even though we can´t determine weather the orthogonals are parallel or not and there for prove the axioms in the projective geometry, our experience (trail and error) tells us that diagonal lines striving against an ideal point represent both depth and distance. If we assume that this empirical experience is valid we should be able to manipulate our perception of space.
By using orthogonal lines that converge the spectator instead of the vanishing point, a reverse perspective, will counteract the feeling of depth and appear more parallel in our perception. The space will look closer and consequently bigger.
This manipulation with perspective is known and a good example of this is the trapezoid shaped entrance to the St. Peters basilica and the massive elliptical square, the ovato tondo which precedes it.
The paradox is that this is not a fundamentally new way to perceive space unknown to the eye. Bernini “camouflaged” the isosceles trapezoid in our visual experience and we end up with more rectangular and parallel impression, something we know and seen before. The same applies for the ovato tondo; they’re no way to decide whether the ellips is actually an ellipse or a circle. The inverse problem applies, again.
Impact on the concept of visual perception?
The Ponzo effect is two lines (A´B´, C´D´), transversals that look different in length in the realm of projective geometry, but have the same length when measured. To achieve or draw this representation and create this "illusion", in reality the two transversals in the object plane have to have different lengths. If they had the same length the result would simply be a logical distribution e.g. as sleepers under a railway track or a ladder.
Given two lengths (AB, CD), there is only at a certain distance in reality when this effect will occur projected and this is rare situation. I other words, the Ponzo effect is a special case in reality without any significance to understand vision, on the contrary. The paradox is that our perception tells us the truth empirically; the second transversal (AB) both looks longer and is actually longer than (CD) even though they in the projection plane, have the same measurements. On the other hand manipulations of the physical space to revoke the laws of perspective can is still be relevant esthetically and architectural. It will not create anything visually new, only uncommon circumstances for something we have seen before.