# Patterns in Multiple Reflections

## Eva Liza Zacarias

### Introduction

The hardest thing when doing a research project, for me, has to be finding the topic you want to do research on. There's so many out there that it's hard to decide which one to choose. The topics are narrowed down to Geometrical Optics when I decided to do something that is math related. After months of decision-making, two possible topics emerge: Caustics and Multiple Reflections. Caustics is a really fascinating topic but as much as I love math, I don't think it's going to be possible for me to understand it in detail. The math involved is really deep and so far, the math that I've learned goes only up to Multivariable calculus. Multiple Reflections, on the other hand, can start of as something simple but if you consider the different cases wherein it occurs, it could be quite complex.

The math involves simple Euclidean geometry since, to a good approximation, light travels in lines as "rays". Some problems can be analyzed in only two dimensions while more complex ones require tracing lines in three dimensions. With only the use ray trace models based on Euclidean geometry and the law of reflection, the project would show how these demonstrations can be understood.

### Reflection Over a Plane Mirror

In Reflections over a plane mirror, the light rays leave a source in all different directions. These rays follows the law of reflection once they hit the mirror surface. To determine where the location of the reflected image, we extend the reflected rays to the other side of the mirror. The point where the extension of the reflected rays intersect would be the location of the reflected image.

### Number of Images Formed:

Think about two mirrors that are at an angle with each other. You would notice that as you move the mirrors closer together, the number of images formed are increasing. When they become parallel, a seemingly infinite number of images are actually produced. According to V. M. Kulkarni Walchand, the number of images formed is not completely independent of the object’s position. A paper by him summarizes the number of images formed in four simple rules.

1. If 180°/q = x, an integer then the number of image formed, N = 2x-1, regardless of where the object is.
2. If 180°/q = x + 0.5, number of images formed , N = 2x when the object lies on the angle bisector and N = 2x+1 on other positions.
3. If 180°/q = x + n/q and n/q < 0.5 then the number of images formed, N = 2x with the location of the object anywhere on the central angular sector of (q-2n)° about the angle bisector. N = 2x+1 when the object lies outside of the given sector.
4. If 180°/q = x + n/q and n/q > 0.5 then the number of images formed, N = 2x + 2 with the location of the object anywhere on the central angular sector of (2n-q)° about the angle bisector. N = 2x + 1 when the object lies outside of the given sector.

Note: Notice that if you take the limit of 180°/q as the angle between the mirrors, q, goes to zero, the number of images formed would go to infinity. This makes sense since that is what we see when the mirrors are parallel.

### The Special Angles

We call these angles "special angles" because these are the angles where a new image appears. When you look at the first of the four rules, you would see that an odd number of images appear when 180°/q = x, an integer regardless of where the object is. The "unpaired" image would be the one in the center. When we solve the equation for q, we get 180°/x = q as the equation for finding those "special angles".

### Finding the Path of the Light Ray

To find an easier way to determine the path of the light ray, we use a method that is similar to "tiling". To do that, you first extend the incident light ray to the other side of the mirror it hits first. Find the reflected image of the mirror that the incident ray did not hit first (mirror A) over the other mirror (mirror B) naming it A'. Then, you find the reflected image of mirror(B) over mirror A' calling it B'. You repeat the same process until the light ray leaves the hinged mirrors. The intersection point of the extended light ray and the mirrors are actually the points where the light rays hit the mirrors. If the intersection point is between the extended light ray and a reflected image of mirror A, then that point corresponds to a point somewhere in mirror A. The similar idea is the same for mirror B. The points are connected in the same order as the arrangement of the extended incident ray hitting the mirrors but the points you are connecting would be the corresponding points in mirrors A and B only.

### Formation of New Images

In the image shown below, the mirrors are perpendicular. Using the same idea as the reflection on a plane mirror, the location of the reflected image can be determined by finding the intersection of the extended reflected ray. In mirrors that are perpendicular however, some of the light rays not only hit the mirrors once but twice. If you extend the second reflected ray, they intersect at another point wherein another image is located.

If the mirrors are perpendicular, the light ray hits the mirrors, at most, twice. When you decrease the angle between the mirrors, it is possible for the light ray to hit the mirror several more times causing more images to appear.

### Fractal Reflections

Fractals are fragmented geometrical shapes that can be divided into parts. Each subdivision appears as a reduced copy of the whole image. Fractals can not only be observed in nature like the systems of blood vessels in our body but also in multiple reflections on curved surfaces.

Fractal reflections occur when silvered globes are stacked close together. The most convenient one being the tetrahedral pattern wherein three of the balls are mounted as a base and the another one is placed on the top. These silvered globes has properties of convex mirrors making the images smaller after each reflection.

### Works Cited:

V. M. Kulkarni. "Number of Images Produced by Multiple Reflection".American Journal of Physics Volume 28 (1960): 317-318

Evans II, Howard E. "Ray Tracing with Hinged Mirrors". THE PHYSICS TEACHER. Volume 34 (1996): 314-315