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. The web provides a viable medium for enhancing mathematical presentations with photo-realistic graphics and video.
Some of the images in this article have associated animations, indicated in the figure caption by the phrase “with linked animation.” Click on the image to view the animation in a new browser window or tab -- these animations are in 1280x720 Quicktime format, so your browser or operating system must be configured (with the Apple Quicktime Player or compatible software) to launch them properly. The animation files are large, so they may take a few minutes to load in your browser.