Patterns in Multiple Reflections
Eva Liza Zacarias
Laser Teaching Center
Department of Physics & Astronomy
Stony Brook University
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.
- If 180°/q = x, an integer then the number of image formed, N = 2x-1,
regardless of where the object is.
- 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.
- 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.
- 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
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.
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
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.
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