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Modern expositors of mathematics, with computers and open source ray tracing software at their disposal, have the tools necessary to create vivid and effective geometric constructions in 3-space. With a little additional effort, animations can be generated, adding the fourth dimension of time. Expository material of this kind is often excluded in ordinary print publication due to the high cost. In contrast, the web provides a viable medium for enhancing mathematical presentations with photo-realistic graphics and video.

The recent videos *Mobius Transformations Revealed* [1] and *Dimensions* [2] are two examples of the use of the open source ray tracer POV-Ray to create wonderfully effective animations for showcasing beautiful mathematical ideas.

The POV-Ray scene description language is well suited for geometric construction. It is our experience that the task of creating effective images and animations is a mathematical activity requiring mathematical thinking, in much the same way as with the classical Euclidean constructions using straightedge and compass.

The purpose of this article is to introduce ray tracing as a tool for mathematical exposition. It will not escape the interested reader that it can also serve as a powerful tool for mathematical exploration and experimentation.

We begin with a gallery of mathematical images to illustrate what can be done using ray tracing. Ten images are shown illustrating various geometric concepts. Six of these images have associated animations, indicated in the figure caption by the phrase “with linked animation.” Click on the image to view the animation (these are in 1280x720 Quicktime format, so your browser or operating system must be configured to launch them properly).

Ray tracers are great tools for creating vivid, photo-realistic images. Properly utilized, they can enrich the mathematical exposition. For example, suppose we wish to define and illustrate the concept of polyhedral duality as it applies to the regular solids. Let's first construct the sphere which circumscribes an icosahedron (Figure 1). The icosahedron intersects the sphere at its 12 vertices.

Figure 1: Icosahedron circumscribed by a sphere.

Planes tangent to the sphere at each of the 12 vertices will bound another polyhedron: the dodecahedron, shown in Figure 2. Polyhedra related in this way are said to be “duals”. Note that the sphere which circumscribes the icosahedron is also inscribed in the dodecahedron, which it touches at the centers of each face. Equivalently, the vertices of the icosahedron mark the centers of the faces of its dual.

Figure 2: The dodecahedron with faces tangent to sphere.

Clearly, this construction can be continued indefinitely at both larger and smaller scales. Just as clearly, both polyhedra have the same symmetry group [3], the Icosahedral group. The full group (including reflections) has order 120 and the rotation subgroup has order 60 [4]. Figure 3 has a linked animation showing the two polyhedra rotating a full turn.

Michael Grady (Southern Utah Univ.), "A Gallery of Ray Tracing for Geometers," *Convergence* (September 2010), DOI:10.4169/loci003350