By Grant Ocean

# 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 (Im2) on the same spot.

You are still 160cm tall and 100cm from M1, i.e., dom = 100cm. And you are 50cm away from M2. The distance between E and M2 is also the distance of the second reflected light (dr2), reflecting off M2 from the incoming first reflected light (dr1), i.e., dr2 = 50cm. Also, the distance between M1 and M2 (dm1m2) is the same as dom, i.e., dm1m2 = dom.

Total distance of the light rays that reflect off you (O) and travel to M1, then M2, and finally reach your eye (E) is: di + dr1 + dr2, where di = dr1. Let’s use a ruler to measure your image on M2 (i.e., Mp = 50cm). Now we can obtain the perceived image height in the second mirror (PIm2) 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 Im2 from the head to the mark on the leg. The result for PIm2 was confirmed; it was about 31.23cm.

As we know from Case A, the distance between the image in the first mirror Im1 and M2 (dIm1m2) is the same as di + dr1, so dIm1m2 = di + dr1 = dIm2, where dIm2 is the distance between M2 and Im2. As a result, there are two possibilities for PIm2. One is that PIm2 could be the perceived image height of the reflection of Im1 in M2, so that Im2 is the reflected image of another image Im1, which is the conventional wisdom. Another possibility is that PIm2 is the perceived image of Im2 projected from E along the extended line of dr2, 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 Im1 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 Im1 of 80cm in height. As mentioned previously, dIm1m2 = di + dr1. Hence, when you measure Im2 on M2 (Mp = 50cm) if reflected from Im1, 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 di + dr1 + dr2. 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, Im1 does not even exist when not observed by an observer.

One could still insist that O has been transformed in whole to Im1 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 = Im1. However, this principle is merely an assumption based on optical geometry, not on the actual measurements. For instance, when measured on the mirror, Im1 = ½ H, as demonstrated in Case B. Moreover, for H = Im1, the reflected light rays have to maintain their intensity from O to Im1. 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 Im1, to M2, and then to E, so that the total distance of dom + dim + dIm1m2 + dr2 is almost twice as much as di + dr1 + dr2. As a consequence, the perceived image in M2, if reflected from Im1, is only approximately half of that when mentally projected from E at the span of di + dr1 + dr2, 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 Im1 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 Im1 will be able to do the same as the original object to have its own light points reconstructed again in M2. The Im1 and Im2 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 dom and dim. 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 Im1 (see the diagram at the beginning of this section) did not affect the perception of Im2. This means that the perception of Im2 has nothing to do with Im1 (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 (dp) 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 Im1 has absolutely no effect on Im2; and there is zero connection between Im1 and Im2. 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 Im2 on M2 and keep all the other values the same. So, we have a new result for PIm2

In comparison to the earlier result for the perceived image height when M2 is 50cm away from the observer, this result shows that PIm2 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 PIm2 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 Ph 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.

# References

Gregory, R.L. (1998). Eye and brain (5th ed.). Oxford: Oxford University Press.

Hollins, M. (1976). Does accommodative micropsia exist? American Journal of Physiology, 89, 443-454.

Kaneko, H, & Uchikawa, K. (1997). Perceived angular size and linear size: The role of binocular disparity and visual surround. Perception 26 (1), 17–27.

Kosslyn, S.M. (1975). Information representation in visual images. Cognitive Psychology, 7, 341-370.

Zygmunt, P. (2001). Perception viewed as an inverse problem. Vision Research, 41(24), 3145-3161.

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