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Mirror Images and Mental Projection

By Grant Ocean

The Inverse-Square Law and the Perceived Size Equation

To begin with, letís get familiarized with a ubiquitous law in the universe, called the inverse-square law, which governs the behaviors of gravitation, electrostatics, acoustics, light and other electromagnetic radiations. The perceived size equation which this paper introduces has been derived from this universal law.

The inverse-square law commonly applies when certain energy, force, and some conserved quantity are evenly radiated outward from a point source in three-dimensional space. Given that the surface area of a sphere (which is 4πr2 in mathematical terms) is proportional to the square of the radius r, as the emitted light (or the reflected light in the case of object perception) travels away from the source, it spreads out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of light (I) passing through any unit area is inversely proportional to the square of the distance (r2) from the point source, positively proportional to the total power (P) of light. Thus, the general equation for the inverse-square law can be written as


   or  

Now we can derive a perceived size equation from the equation above. The total size of an object (S) corresponds to P, which can be thought of as the total photons reflecting off the object. The perceived size (Ps) of the object corresponds to I, which could be the amount of photons entering the eye. The distance (d) between the observer and the object is the same as r. As a result, the perceived size equation can be expressed as follows


However, the perceived size of an object is always measured by an active observer. The point where one places the ruler to measure the object will affect the perception of the objectís size. Therefore, we need to add a term, the measure-point (Mp) to the equation to resemble how the perceived size of an object is actually measured and materialized in the human situation. The measure-point is where the ruler is placed away from the eye (or the nodal point to be precise), so that Mp is the value of the distance between the eye and the ruler. The measure-point is positively proportional to the perceived size. That is, placing the ruler farther away from the eye or closer to the object leads to a larger perceived size. Consequently, the perceived size equation is re-written in the form below


       
       

We can see that the perceived size (Ps) is determined by the measure-point (Mp) when the objectís size (S) and the distance (d) are constant. So Mp plays an important role in our size perception. As a matter of fact, we do not need to use a ruler at Mp to influence how we perceive the size of an object. We can replace the ruler with any object so long as we focus on the object. For instance, if you hold up your thumb in front of you and focus on it, the objects beyond it appear smaller when you move your thumb closer to your eye, and vice versa. In this case, Mp is the focal distance. The larger value of Mp results in a larger perceived size, and vice versa. As for the general inverse-square law I = P/r2, Mp could be regarded as an intensity detector. The closer the detector is placed towards the source, the higher intensity will be measured. In so doing, such a detector is acting the same way as the ruler.

        To link our perception with a universal law is a new approach to how our mind functions. Our brain or mind is envisioned to behave like all the other entities in the world to obey the universal laws. Instead of treating the size perception as the geometrical relationship between objects and our eyes, the new approach envisages the objectsí sizes we perceive as being equivalent to various magnitudes of energy intensity following the inverse-square law. This approach might sound far-fetched according to our conventional viewpoint. But, its validity and appropriateness should be judged by the actual experiments, in particular by how well the perceived size equation can explain and predict the mirror images.

The objectís size (S) is the total area of an object in two dimensions, i.e., vertical and horizontal dimensions. If both dimensions of an object are doubled, the total area of the object will increase by fourfold. When the vertical and horizontal dimensions of the object are tripled, the total area of the object grows by nine times. As a result, the objectís size increases in the same proportion as the square of distance, i.e., . If only one dimension of the object is concerned, i.e., the linear size of the object such as its height, the objectís height (H) changes in the same proportion as the distance (d), i.e., . Thus, the perceived size equation can be simplified as


where Ph is the perceived height. This simpler version of the perceived height equation is not only easier to be calculated, but more importantly the values of the terms in the equation are also much easier to be measured; so it has more practical usage. Therefore, this simplified equation, instead of the original perceived size equation, will be used for the actual calculations and follow-up experiments throughout the paper. The perceived height equation and the perceived size equation are interchangeable in meaning in this paper; and the perceived height will be sometimes referred to as the perceived size.

It is very simple for you to verify the above equation. All you need are a measuring tape and a transparent ruler. Place the measuring tape alongside the edge of a table and set the zero point at one end of the table. Secure one eye of yours at this end with a chin spporter and above the zero point of the tape, blind-folding the other eye. Now find an object (e.g., a book), measure its height (H), and put it at certain distance (d). Set the ruler at a distance (Mp) close to the eye and measure the height of the book. This measurement is the value of Ph. Then plug all the measured values in the equation and determine whether the equation is balanced. If the equation is balanced, it means that it works in the real world and can adequately tell us about the workings of size perception. The accuracy and efficacy of the equation will be tested again and again in the cases given below.


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