### 6. Conclusion

Once a scene has been created in a ray tracer, there are an infinite variety of possible viewpoints and transformations to try out. There is not just one image, but a virtual world that can be explored via animation. How many of us have built or are willing to undertake the painstaking effort of building a trihedral kaleidoscope? With a ray tracer, you can construct one virtually and experience the extraordinary effect of seeing a truncated dodecahedron formed in space from the reflections of a single cylinder. This is an excellent way to develop the mathematical imagination.

We have just barely scratched the surface of what is possible for the imaginative expositor, student or explorer of mathematics who uses a ray tracer. A look at Coxeter's beautiful book: *Introduction to Geometry* [1] or Needham's remarkable text *Visual Complex Analysis* [2] should suggest a wealth of fascinating new experiments.

### 7. References

[1] Coxeter, H.S.M., *Introduction to Geometry, *2nd edition (1989: Wiley).

[2] Needham, Tristan, *Visual Complex Analysis *(1997: Oxford Univ. Press), p.140-142 http://www.usfca.edu/vca/.

[3] Wikipedia contributors, "Dodecahedron," *Wikipedia, The Free Encyclopedia* (October 2009) http://en.wikipedia.org/wiki/Dodecahedron.

### 8. POV-Ray Related Links:

- Wikipedia article on POV-Ray: en.wikipedia.org/wiki/POV-Ray
- Official POV-Ray Tutorial: www.povray.org/download
- Official POV-Ray Reference Manual: www.povray.org/download
- Lohmueller has a set of excellent tutorials illustrating the use of POV-Ray to teach Analytic Geometry: http://www.f-lohmueller.de/pov_tut/a_geo/a_geo__e.htm

### 9. Footnote for Mac Users

The Mac version of POV-Ray at povray.org was compiled in 2004 for the PowerPC. If your system is Intel based and you encounter difficulties running the package, there are two options. The first is to use Mega-POV instead (megapov.inetart.net). This is a custom and unofficial patched version of POV-Ray with a graphical editor and many useful extensions. The other option, recommended for those who prefer the command line, is to download the Unix source and compile it yourself.