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        3rd reflection image from object A.  Ray goes from mirror 1 to 2 to 3.            

Theoretically, the number of reflection images is our kaleidoscope is infinite. We conclude our example with the images resulting from three reflections.

As before, we begin with the object in the center. Consider two light rays that begin at the object and reflect off the bottom mirror.

After reflecting from the base mirror, the rays move upward to the upper right hand side mirror. The ray pair next heads towards the upper left mirror and reflects a third time.

The resultant rays are extended behind the upper left mirror with the virtual image located at their intersection.

       

We continue with the object in the lower left hand corner of the original triangle. The ray pair goes from the object to the bottom mirror where the two are directed upward. Both rays reflect off the upper right mirror in the direction of the upper left mirror.

These rays diverge after their third reflection, so the image is virtual.

            3rd reflection image from object B.  Ray goes from mirror 1 to 2 to 3.        
        3rd reflection image from object C.  Ray goes from mirror 1 to 2 to 3.            

Rays that go from the object to the bottom mirror, to the upper right mirror, and over to the upper left mirror are used to find the third object. The corresponding ray diagram is on the left.

The results obtained for third reflection images to this point are color coded with the first reflection mirror and summarized in the schematic.

            Summary - third reflection images for rays from mirror 1 to 2 to 3.        
        3rd reflection image from object A.  Ray goes from mirror 1 to 3 to 2.            

A second set of third reflection images form due to light following paths that start at the object, reflect on the base mirror, the upper left and then upper right mirror.

As the reader might expect, the images formed by these ray sets are located to the upper right of the original object trio.

If you are interested in drawing out these ray diagrams for yourself, it would be a good idea to use graph paper and a good protractor.

(I employed a bit of algebra to be certain of the angles that appear in the sketches.)

            3rd reflection image from object B.  Ray goes from mirror 1 to 3 to 2.        
        3rd reflection image from object C.  Ray goes from mirror 1 to 3 to 2.            

With that said, it is important to remember that we have relied solely on the principle that light bounces off the reflective surfaces like a ball does when hitting a hard surface.

The physics behind beautiful kaleidoscope images is remarkably simple!

The second set of third reflection images is found in the graphic to the right. The bottom mirror, where the first reflection occurs, is shaded in red as are the images.

            Summary - third reflection images for rays from mirror 1 to 3 to 2.        
        3rd reflection image from object A.  Ray goes from mirror 1 to 2 to 1.                

A third set of unique images can be obtained from yet another ray set. Here, the light begins at the object, reflects from the lower and the upper right mirror before returning to the lower mirror where the third reflection occurs.

While there are other possible light ray sets, these three provide a unique set of images. Due to the three fold symmetry there are other ways to set up the ray diagrams, this being one possibility.

   

The images predicted for the flat mirrors are virtual because real light rays do not merge to form the image. Rather, we perceive the image to be at the location where such light rays could have come from.

                3rd reflection image from object B.  Ray goes from mirror 1 to 2 to 1.            
3rd reflection image from object C.  Ray goes from mirror 1 to 2 to 1.                

OK. This is the last ray diagram (cue hallelujah chorus).

The final three third reflection images are put together in this graphic.

                Summary - third reflection images for rays from mirror 1 to 2 to 1.

Now it's time to put the third reflection image sets together with the original object group.

Summary - unique third reflection image sets for rays from objects to mirror 1.

You can repeat our ray diagram exercise above, use symmetry (my choice), or trust me to obtain the images from ray paths that have a first reflection at the upper right and left mirrors. In any event, you should obtain an image template like the one below.

The original objects are in black, first first reflection images in gray, and second reflection images in light gray. Third reflection images and mirrors are colored to match their first light reflection.

Color coded results for thrid reflection kaleidoscope images.

Superimposing this template on the expected kaleidoscope image, one obtains the picture below.

3rd reflection template overlaid on kaleidoscope image.

Finally, the template is removed. The kaleidoscope prediction below represents the original object, the three sets of first reflection images, the six second image sets, and the nine third reflection image groups.

Original, first, second, and third reflection kaleidoscope images.

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