Second Reflection Images

The images considered on the previous page are the common reflections seen in a single mirror. It is the presence of another mirror that produces the infinity of images that are the hallmark of the kaleidoscope.

Consider the two light rays in the sketch below. The light originates at the center object and moves toward the bottom mirror. As we observed previously, the light reflects at the mirror's surface. But, the light paths do not end with a single reflection.

After moving away from the bottom mirror, the light reflects off the top right mirror. This second reflection gives rise to a second virtual image.

The outgoing light rays diverge. We extend them behind the mirror to locate the virtual image, marked at the intersection of the ray extensions.

2nd reflection image of object A. Ray hits mirror 1 and then mirror 2.

The bottom left hand corner object produces an image from a bottom to upper right mirror reflection pair. This image is located via the ray diagram to the immediate right.

2nd reflection image of object B. Ray hits mirror 1 and then mirror 2.

Restricting ourselves to light paths that begin at the object and subsequently reflect from the lower to the top right mirrors, we locate our third object's virtual image. The rays used for this are shown in the figure below.

2nd reflection image of object C. Ray hits mirror 1 and then mirror 2.

We summarize the findings from the bottom to right reflection pairs in the following schematic.

Summary of first image set from 2nd reflection, mirror 1 to 2.

Our next step is to consider the bottom to upper left mirror reflections. As before we start with the center object.

By symmetry we anticipate that the image would be found to the left of the object, as demonstrated in the ray diagram.

2nd reflection image of object B. Ray hits mirror 1 and then mirror 2.

The lower left hand object has an image that appears on the left as well. The precise location of the image is best found by drawing a picture.

This second set of second reflection images is nearly complete. The image location of the circle object in the upper section of the triangle is determined in the ray diagram that follows.

2nd reflection image of object C. Ray hits mirror 1 and then mirror 2.

A summary of the images produced by the bottom to upper left mirror reflections is provided below.

Summary of second image set from 2nd reflection, mirror 1 to 2.

It is convenient to summarize all of the second mirror reflection images with first reflections at the bottom mirror.

The bottom mirror and and the corresponding second reflection virtual images are shaded in red.

Color coded summary. First reflection mirror and second reflection images are in red.

One can apply the procedure presented above for the bottom mirror second reflection images two more times to obtain the full set of second reflection kaleidoscope images. We present the result of this exercise below.

2nd reflection images. Shading matches images with the first mirror reflection.

Note that original objects and first reflection images appear in black. The bottom mirror corresponding second reflection images obtained above are shaded in red.

Blue and yellow-green are used to designate the upper left and right mirrors with their image sets.

This object-image template is used to create the actual kaleidoscope picture. They are shown together in the next figure.

1st and 2nd reflection image orientations.

Finally, we eliminate the template to reveal the anticipated kaleidoscope image produced by the object and first and second reflection images.

Summary of images from 1st and 2nd reflections.

Third reflection images will complete a hexagon, the pattern that is familiar to those who have looked through a 60°-60°-60° kaleidoscope.

Go to third reflection images.

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