Reflections in spheres

Reflections in 4 spheres

I love mirrors. They can create some very interesting mathematics. It is therefore surprising that although I have been working with raytracers (which do reflection very well) for quite a long time that I have never used them to look at reflections. I have finally dealt with that omission.

The image above looks at four spheres placed at the corners of a tetrahedron so that they are nearly touching. Without the mirror finish they would look like this:

The four spheres without reflection

In order to get more pictures, and to help understand the pattern of reflections I now started to colour the surroundings. As we look at the spheres our gaze is bounced around until it eventually heads away from the spheres again. The question is, at that stage what direction is our view going. In other words, what are we actually looking at within all the reflections? To show this I created the set up below:

Reflection set up

As you can see the spheres are in a yellow bucket with a blue bottom and red top. The position of the camera is shown with a black cuboid and the light is the white sphere.  The red top will be pushed over the model for the images and the light switched on! Note that I place the light behind the model so we would not see it in direct reflection. So what do we get?

Up, level and down colours

Now we can see whether we are really looking up (in red) level (yellow) or down when we look at the reflections. Can you look at the image and work out more about the paths various lines will take before heading out? The beautiful structure comes in as there are paths that take a very long, or even infinite time to leave the spheres.

A second image looks for right and left:

Left (Blue) and Right (Red)

Finally just for fun, an image where the back sphere is glowing so you can see how it is reflected around the front three spheres:

Glowing back sphere

For more on the mathematics of sphere and circle reflections, and how they can be generalised to whole geometric worlds (with great images) see Indra’s Pearls, you can also use the blender file for these scenes.