Case C: Images in the Multiple Mirrors

The diagram below depicts a situation where you stand in front of the first mirror (M₁) and look into the second mirror (M₂). Your position in the diagram is O/E, where you stand as an object O and use your eye E to observe your image in the second mirror (I_m2) on the same spot.  

You are still 160cm tall and 100cm from M₁, i.e., d_om = 100cm. And you are 50cm away from M₂. The distance between E and M₂ is also the distance of the second reflected light (d_r2), reflecting off M₂ from the incoming first reflected light (d_r1), i.e., d_r2 = 50cm. Also, the distance between M₁ and M₂ (d_m1m2) is the same as d_om, i.e., d_m1m2 = d_om.

Total distance of the light rays that reflect off you (O) and travel to M₁, then M₂, and finally reach your eye (E) is: d_i + d_r1 + d_r2, where d_i = d_r1. Let’s use a ruler to measure your image on M₂ (i.e., M_p = 50cm). Now we can obtain the perceived image height in the second mirror (P_Im2) as follows

A follow-up experiment was conducted to confirm the calculations. Somebody was measured from the tip of the head and left a visible mark somewhere on the leg which was 160cm from the head (because it is hard to find someone who is exactly 160cm tall). The person stood on the O/E spot, which was 100cm from M₁ and 50cm from M₂, with the eye secured over the spot. A ruler was placed against M₂ to measure I_m2 from the head to the mark on the leg. The result for P_Im2 was confirmed; it was about 31.23cm.

As we know from Case A, the distance between the image in the first mirror I_m1 and M₂ (d_Im1m2) is the same as d_{i +} d_r1, so d_{Im1m2 =} d_i + d_r1 = d_Im2, where d_Im2 is the distance between M₂ and I_m2. As a result, there are two possibilities for P_Im2. One is that P_Im2 could be the perceived image height of the reflection of I_m1 in M₂, so that I_m2 is the reflected image of another image I_m1, which is the conventional wisdom. Another possibility is that P_Im2 is the perceived image of I_m2 projected from E along the extended line of d_r2, which is the conception of the new account.

From Case B, we know that you only need a mirror that is one-half of your height to view your full image. We can make the first mirror 80cm in height. So you see the full reflection of your body (160cm in height) in this 80cm high mirror. Now you turn to the right and see the reflection of I_m1 in M₂. The image in the first mirror is only 80cm instead of 160cm, limited by the mirror size. Therefore, the reflection in M₂ can only be I_m1 of 80cm in height. As mentioned previously, d_{Im1m2 =} d_i + d_r1. Hence, when you measure I_m2 on M₂ (M_p = 50cm) if reflected from I_m1, the result will be

This outcome shows that the perceived image height in M₂ is merely a half of that when mentally projected from E at a length of d_i + d_r1 + d_r2. Since the experiment has confirmed that the perceived image height in M₂ is 31.23cm while measured against M₂, not 15.62cm calculated above, we come to an important conclusion that the image in the first mirror is not reflected in the second mirror. Consequently, I_m1 does not even exist when not observed by an observer.

One could still insist that O has been transformed in whole to I_m1 since it is after all a geometrical principle that the object’s height is the same as the image height in the first mirror, i.e., H = I_m1. However, this principle is merely an assumption based on optical geometry, not on the actual measurements. For instance, when measured on the mirror, I_m1 = ½ H, as demonstrated in Case B. Moreover, for H = I_m1, the reflected light rays have to maintain their intensity from O to I_m1. As we know, the light intensity follows the inverse-square law; as such, the intensity goes down when the distance lengthens. Let’s assume that the light rays could travel to locations behind mirrors. The light has to travel from the source O, to I_m1, to M₂, and then to E, so that the total distance of d_om + d_im + d_Im1m2 + d_r2 is almost twice as much as d_i + d_r1 + d_r2. As a consequence, the perceived image in M₂, if reflected from I_m1, is only approximately half of that when mentally projected from E at the span of d_i + d_r1 + d_r2, which is then again confirmed by the experiment. This is intended to show those, who believe that they do really see the duplicated images in the mirrors, that it is equally impossible for either the image on the mirror or the image at a location behind the mirror to be reflected in the second mirror.

By the convention, it is believed that I_m1 is physically reconstructed point by point from the light points reflecting off the original object at a location that is straight across the object and has the same distance from M₁ as the object. This physically reconstructed I_m1 will be able to do the same as the original object to have its own light points reconstructed again in M₂. The I_m1 and I_m2 are believed to have physical existence. However, our calculations and experiments, especially the following experiment, provide the convincing evidence that the mirror images do not have physical existence and they depend on the mind for their existence.

The diagram below is a copy of the diagram in Case A. The only difference is that an opaque panel (represented by a solid bar on the mirror in the diagram) is placed on the mirror between O and I, along the line of d_om and d_im. The opaque panel essentially eliminates any possibility for O to be reflected across the mirror at I, because the connection between O and I is completely cut off. The same experiment in Case A was conducted again by using the same 20cm high book put at 50cm from M and 50cm from E. But this time, a panel bigger than the book was placed on the mirror between O and I, as shown in the diagram below. After doing so, an observer at E could still perceive the mirror image I; and the perceived image height was still measured 2cm when the ruler was positioned at 11.18cm from the eye. This experimental result indicates that the mirror image I we perceive can only be the projected image out of the mind given that the reflection of O has been rendered unattainable by the blockage of the opaque panel. Likewise, a panel placed between O and I_m1 (see the diagram at the beginning of this section) did not affect the perception of I_m2. This means that the perception of I_m2 has nothing to do with I_m1 (which has not been realized and so does not exist). Accordingly, it must be concluded that the image we perceive in the second mirror is none other than the projected brain image. Now it is clear that there is no reflecting line between O and I at all; and the only connecting line should be the line of projection (d_p) between the final mirror image and the observer’s eye.

By now, we have solved the quandary encountered in Case A, i.e., whether the mirror images are the reflections of the light points or they are the projections by the mind. At this point, it should be obvious to anyone who has carefully examined the calculations and experiments that I_m1 has absolutely no effect on I_m2; and there is zero connection between I_m1 and I_m2. They both are the projected brain images. As such, they are both connected to the mind.

In sum, there is no connection between objects and mirror images, only the eye (or the mind) and the mirror image. The mirror image is not reflected, but projected. The word “reflection” is a misnomer when it is applied to mirror image formation. As we know now, there is no reflection, just projection. We have to change the concept of reflected image to projected image based on our new understanding.

Now let’s move the second mirror closer from 50cm away to 10cm away from the eye. We still measure I_m2 on M₂ and keep all the other values the same. So, we have a new result for P_Im2

In comparison to the earlier result for the perceived image height when M₂ is 50cm away from the observer, this result shows that P_Im2 has reduced from 31.23cm to 7.61cm when M₂ has been moved from 50cm away to 10cm away from E. The experiments confirm that the closer M₂ is moved to E, the smaller P_Im2 becomes, and vice versa. These results are similar to those of the moving windows in Case B. The closer the window is from you, the smaller the perceived height P_h is. But, in the moving window case the distance has not changed. In contrast, in this case the total distance between O and E has shortened from 256.16cm when M₂ is 50cm away to 210.25cm when M₂ is 10cm apart. As a consequence, the closer an image is to the eye, the smaller it appears.

These results, more than the moving window case, violently contradict our conventional conception of size perception. The visual angle is supposed to be determined by the distance. The shorter the distance between O and E is, the larger θ is; and vice versa. This is a simple geometrical fact that the angle of a vertex will increase as the adjacent line decreases when the opposite line stays the same, and vice versa. However, the new account of size perception and application of its equation have shown that a closer mirror or shorter distance could, under the special circumstances, lead to smaller perceived size.

Therefore, the conventional accounts of size perception such as the visual angle, SDIH or the relatively new perceived visual angle (Kaneko & Uchikawa, 1997) and the new account are totally incompatible. The acceptance of one account has to be done at the expense of abandoning the others.