Visualizing the fourth dimension using geometry and light

Techniques for visualizing mathematical objects in four-dimensional (4D) space that exploit four-dimensional lighting effects are explored. The geometry of image production, stereography, and shadows in 4D is analyzed. Alternatives for smooth and specular shaded rendering of curves, surfaces, and solids in 4D are examined and a new approach that systematically converts curves or surfaces into uniquely renderable solids in 4D space by attaching spheres or circles to each point is proposed. Analogs of 3D shading methods are used to produce volume renderings that distinguish objects whose 3D projections from 4D are identical. Analyzing the procedures needed to justify and evaluate a system as this for teaching humans to `see´ in four dimensions leads to the proposal of a generally applicable four-step visualization paradigm