The 2-mirror system is the most common system used in kaleidoscopes.
In the 2-mirror system the angle between the two mirror surfaces determines the number reflections that make up the image.
The smaller the angle the higher number of reflections, those reflections are called folds. When two folds meet, they form a point.
It is difficult to create this mirror system in real life if you don't have two mirrors and a designated space between the mirrors that produce known angles.
Fig.1 - Two Mirror System
To make this easier there is an online physics applet we can use to simulate a 2-mirror system.
In this applet we can manipulate the angels between the mirrors and find the number of reflections or folds present when changing the angle of the mirror.
Using this applet to find the number of folds at different angles, we can derive a relationship between the angle between the mirrors and number of folds formed.
See table two for the number of folds create as the angle is change.
Using this table, we can see that folds are determined by dividing 360° by the angle between the two mirrors and subtracting one.
Points are determined by dividing 180° by the angle between the mirrors.
As the angle decreases the number of folds increase. The mirrors 1 and 2 are at some angle, θ, and 2’ is the image of mirror 2 through mirror 1.
Also, 1’ is the mirror image of mirror 1 through mirror 2.
Mirror 2’’ is the mirror image of mirror 2 through the imaged mirror 1’ and mirror 1’’ is the mirror image of mirror 1 through imaged mirror 2’.
This continues to the other end throughout 360 degrees.
When an object is placed between the two mirrors there are images through mirror 1 and mirror 2 and mirror images of these images through 1’ and 2’.
Also, remember the mirror images have their own images in the mirrors 1’’ and 2’’ and so on until the point where two images get overlapped or their rays get overlapped.
The images only overlap when the angle at the top is equal to θ and the images do not form when the angle at the top is less than θ.
So, θ has to be less than 360°, n(θ)+α=360, where α is the angle at the top is less than θ.
If α= θ, it means that 360 is a multiple of θ, then (n+1) θ=360, and n+1=(360/ θ), where n+1 is equal to number of segments the whole 360 degrees is divided by the images of the mirrors.
In each of the segments only one image formed is real so the actual number of images is (n+1)-1=n, so n=(360/ θ)-1 (8).
Three mirror system has more complicated math but it all boils down to reflections and reflactions of reflections. Let's take a 60-60-60 degree triangle mirror system
and look at the resulting reflection. The first reflection is the reflection of the objects. The second reflections are the images formed of the reflections of the
reflections. The last images formed are reflections of the refelctions reflection. I made the following Geogebra applet that deminstrates these reflections
and forms some fun kaleidoscope images.
