According to the simplified perceived height equation, we are able to obtain the accurate value of the perceived height of an object once we know the exact height of the object, its distance from the observerís eye and the measure-point. The traditional geometrical account is concerned predominantly about how an object (or a mirror image) falls on the retina to subtend a certain visual angle which in turn affects our size perception; as such, the geometrical rendering of size perception is always about the geometric connection between the object and the eye. In contrast, the new account of size perception, based on the inverse-square law and the derived perceived size equation from it, pays more attention to the path of the light rather than simply the space between the object and the eye to determine the distance. In so doing, the distance between the object and the eye is considered to be the total traveled distance of the light that has started reflecting off the object and ends up striking the eye. As for the plane mirror, shown by the diagram below, the total distance traveled by the light from the object (O) to the eye (E) is the combination of two parts: the incident distance (di) and the reflected distance (dr). Accordingly, the perceived height of the image in the plane mirror is the product of the objectís actual height over di plus dr, modified by the measure-point.
we do not perceive the object as being located at O even though the reflected light has originated from there and
travels to E by the way of di and dr. Rather, we perceive the object as being located at I as an image behind the mirror. The
explanation offered by the present paper for this phenomenon is that our mind projects
the image of the object along the extended line of dr to a location that is the same distance as that
between O and E (which is di
The above diagram shows an object O and an observerís eye E in front of a plane mirror M and its image I behind the mirror, and all the relevant distances. Now we are going to assign definite values to these distances and relevant terms in the perceived size equation in order to calculate the perceived size, and conduct experiments to verify the calculations. The height of O is 20cm (i.e., H = 20cm). The distance between O and M is 50cm (i.e., dom = 50cm). The distance between O and E is 50cm (i.e., doe = 50cm). The distance between E and M is 50cm. In addition, as mentioned earlier, di is the distance of the incident light from O to M and dr is the distance of the reflected light from M to E. The length of di can be obtained by the formula:
Next, an experiment was conducted for the perceived image in the plane mirror according to the above diagram and the assigned values. The same book with 20cm in height was put at 50cm from a plane mirror. The observerís eye was secured at a spot which was 50cm from the book and also 50cm from the mirror. A ruler was placed at 11.18cm from the eye and on the line of sight. The measurement of the book image, which is the perceived image height (Pih), was 2cm in height as well. Thus, we know for sure that Pih = Ph when the total distance, the actual height of the object and the measure-point are the same. By and large we perceive the images in the plane mirror the same as the ordinary objects in terms of the perception of their sizes.
Assuming dom = dim as believed by the conventional
conception, the distance between E
and I, which can also be called the distance
of projection dp (if the
mental projection thesis were true), is calculated as below
If the principles that dom = dim, and O = I have been justified by sound observations and reasoning, then the perceived size equation is proven to be quite accurate since it is in good agreement with the principles. As long as these principles are correct, the perceived size equation is correct as well.
The conventional account of optical geometry assumes that there is a one-to-one relationship between O and I. This is also a commonsense understanding. When you look into a mirror, your image in it will stare right back at you. It seems that there is a direct connection between yourself and your image in the mirror. Every point of your image in the mirror can be conceived as connecting to each corresponding point on yourself with a straight and invisible line. The fact that O and I are directly across each other and have the same size and shape at the same distance from the mirror further confirms the belief that the mirror images are copied or reconstructed point by point along those point lines. This is the conventional account of mirror reflection.
However, the new account of size perception leads to a new conception of mirror reflection. The above calculations and experiments, i.e., Pih = Ph and dp = di + dr, demonstrate that, rather than reflecting directly across the mirror, the mirror image could be the product of mental projection along the extended line of the reflected light. Our mind takes in information (which is the magnitudes of light intensity) from the photons entering the eye and forms brain images (Bi) based on the perceived size (Ps) of the mirror images as far as the size/shape perception is concerned. On the other hand, Ps of the mirror image is the outcome of mental projection of Bi. Without Bi, Ps of the mirror image cannot be projected out; likewise, without Ps, Bi cannot be formed. Hence, Bi and Ps are coexistent and interdependent. It is impossible for one to exist without the other, i.e., Bi = Ps. The formation of Ps and the projection of Bi of the mirror image occur simultaneously. It means that it takes no time for mirror images to appear. The mind assumes that the reflected light is the direction in which the light source has originated. The mind then projects Bi back to the source in accordance to the total distance of the incident and reflected light.
The mind must have a way to know the distance between the eye and the source besides knowing the intensity of the light, just like the planets in the Keplerís third law which are aware of their exact distances from the Sun. The law states that the square of the orbital period of a planet is proportional to the cube of the mean distance from the Sun. Without the intrinsic knowledge of the mean distance from the Sun, it is unimaginable that the planets of different masses could orbit the Sun within a time frame precisely related to the distance.
It is nearly impossible to know how the mind and planets could ascertain the definite distances from other objects. But we can proffer a conjecture that the mind might rely on the photon disparity to determine distances from objects. Letís assume that the photons A and B were perfectly close together when their journey started at the source; and they are certain magnitude apart after entering the eye. The mind may be able to quantify how much distance the photons A and B have traveled on the basis of the disparity of these two photons. Nevertheless, the distance information must have been built in the inverse-square law itself.
Based on the facts that dp = di + dr and O = I, we can speculate that the motivation for the mental projection is to restore the original objectís size/shape and location. The mind definitely has the precise information about the distances from objects no matter how it obtains them. The mind projects out Bi / Ps to a location that is exactly the same distance as that of the source to restore its actual size by the means of this formula: H = Ph∑ d/Mp, where the values of Ph, d, and Mp are all somehow known by the mind.
Besides whether this kind of motivation exists or not, it is obvious that the mirror imageís size is the same as the objectís size (i.e., Hi = H). More importantly, the distance between the mirror image and the eye is exactly the same distance from the eye to the object. This might be a strong evident to support the stance that mirror images are mental projections. On the other hand, the proponents of optical geometry may argue that the same distance from the mirror for both O and I and the equality of O and I also provide solid evidence to suggest that mirror image is an exact reconstruction of O behind the mirror. To uphold the mental projection notion, one can counter-argue that it is simply a geometrical coincedence that dom = dim, resulting from the mental projection with a purpose to return to the original location.
The optical and projected accounts
are equally plausible thus far. It is a draw between these two different
accounts regarding how the mirror images are formed, whether by light rays or
by the mind. Neither side has the convincing evidence to prove the other side
completely wrong. The tiebreaker will be offered later on in Case C.
Notwithstanding, it is worth reiterating the view of the paper that the mirror image is not the representational image in the brain, but is the mental image projected out of the mind. The brain image and the perceived size are interchangeable and can both be regarded as percepts as far as the objectís size is concerned. The mirror image exists in the mind and at the same time also in the external world.