Math + Making

A student blog for Math 189AH: Making Mathematics at Harvey Mudd College

Coxeter Group Kaleidescopes and Stereographic Projection

Adam Sage
Tia Tounesi

A Kaleidoscope is an example of a Coxeter group, a tessellation shape generated by reflections in a mirror. The Kaleidoscope we made has the fundamental shape of an equilateral triangle, one of the simplest examples of a Coxeter polygon, since its interior angles are π/2 degrees each. A few other types of triangles would have worked too though, such as an isosceles right triangle, or a 30-60-90 triangle. These shapes are shown to generate non-overlapping, tiled shapes via the Gauss-Bonnet Theorem (link here for more information).

For our specific Kaleidoscopes, we wanted to improve slightly on the methods that were used before. We did this using cardboard in order to generate a more rigid structure for the reflective surface to sit on. This leads to a structurally sound shape, but unfortunately the corrugated surfaces of the cardboard leaves some small bumps which cause the overall shape to be less cleanly reflective.

For this project, we made lampshades project a shadow of an image of our choice. The method we’ve applied to do this is called stereographic projection, which is the same method used by cartographers to draw maps of the Earth and Sky. The shadow projected is a 2D map of the 3D sphere. The conventional way of stereographic projection is starting with a pattern on a sphere and producing a flat map, but by working backwards and with the help of code and blender, the 2D image can be turned into a sphere. This way, the light shining through the sphere will project a shadow of the original 2D image.

What’s interesting is that, like a map, the shadow’s projection distorts the image’s size but the angles on the sphere are exactly the same in the shadow.

We changed a few things this time around that produced different results than the project last week. A mesh was generated of the images we chose, and when it came down to printing, we implemented a few changes that increased the shadow’s resolution. Each line segment forms the side of a triangular mesh, and we found that increasing the sampling makes the image less pixelated. Making the sphere’s shell thinner also made the image in the shadow much clearer. However, we also added tree supports when 3d printing instead of grid supports, and this made it much harder to remove the supports and cleaning the print became much more tedious than when the supports were grid supports.



Leave a Reply

Your email address will not be published. Required fields are marked *