Images in a Pair of Parallel Mirrors

It is traditionally believed that the image I₁ is the image resulting from the reflection of the object O across the mirror M₁, as shown in the diagram below. The image I₂ is an image of the image I₁, found by reflecting the image I₁ (which essentially becomes an object on its own) across the mirror M₂. This process could continue indefinitely, producing images of images for an infinite number of images. The locating of images is an extension of the principle that the image distance to the mirror is the same as the object distance to the mirror. In short, the images of an object will reflect themselves back and forth following d_om = d_im rule.

As we have already known from the last section, the calculations and experiments do not support the conventional account that one mirror image could be reflected from another mirror image. The mirror images simply do not exist without an observer’s perceiving them. Based on the new conception of mirror image formation, i.e., that they are mental projections, a thesis of multiple projections is proposed for the images in the parallel mirrors.

As shown in the diagram above, the first mirror image I₁ is projected by the eye E following the first reflected light (1L_r) which reflects from the first incident light (1L_i). The perceived height for I₁ can be calculated the same way as in Case A. The second mirror image I₂ is projected by E along 2L_r3 which is reflecting from 2L_r2, 2L_r1, and 2L_i respectively, where 2L_i is the second incident light reflecting off the object O; 2L_r1 is the first reflected second light reflecting off M₁ from 2L_i; 2L_r2 is the second reflected second light reflecting off M₂ from 2L_r1; and 2L_r3 is the third reflected second light reflecting off M₁ from 2L_r2. The eye projects out I₂ along the extended line of 2L_r3. The total length of the projection is the addition of 2L_i, 2L_r1, 2L_r2, and 2L_r3, which are all distances. Thus, the perceived height for I₂ can be calculated as follows

In the similar manner, we can calculate any image insofar as we know how many of the reflected light rays are involved, such as the perceived height for I_n:

It should be pointed out that the above diagram is a simplified version of how an object will appear in the parallel mirrors. The diagram only shows the images of the front side of the object. The back side of the object will hit and reflect off M₂ first, then hit and reflect off M₁ and enter the eye. As such, there should be a pair of images at each location in M₁ if you are looking at M₁.

The perceived size equation when applied for multiple projections in parallel mirrors has two important implications. First of all, our mind has to project images faster than the speed of light. In other words, the mind must project out a pair of images before the next pair of the reflected light rays enters the eye. It is similar to the situation where you race with the light and you always reach the finish line ahead of the light. The conclusion we have to make from this fact is that the multiple projections of images cannot be physical in nature because nothing that is physical in this world can travel faster than light. Consequently, the image projections must be mental. Whether or not the mental content of the mind stems from the physical processes in the brain is not the concern here. We simply take the mind as a non-physical entity to fit the fact that the image projection is faster than light.

Secondly, the denominator part of the perceived size equation for parallel mirrors will have more distance terms added each time an additional image is projected. It could have an infinite number of terms added and become infinitely long since the parallel mirrors could possibly have an infinite number of images in them. The implication of this analysis is that it is impossible for “unconscious inference” to take place as far as the size and location of objects are concerned. Let’s assume that our brain could infer the size and distance for I₁ at a lightning speed. As we know, the mental operations take a finite time to accomplish (Kosslyn, 1975). The more complicated the operation is, the longer it takes to accomplish. By logic, the inference will take a slightly longer time for each additional image. No matter how infinitesimal the slow-down is, eventually it will take an extremely long time for the inference when the number of projected images nears infinity. Therefore, the faster than light image projections do not have any temporal room for any kind of “unconscious inference”. The possibility of the perceived distance, which is equivalent to the inferred or judged distance, in the traditional SDIH has been excluded due to this analysis.