Filling Space
The torus plays a central role in the Hopf fibration of the 3-sphere, S3, over the ordinary sphere, S2, which has circles, S1, as fibers. When the 3-sphere is mapped to Euclidean 3-space by stereographic projection, the inverse image of a circle of latitude on S2 under the fiber map is a torus, and the fibers themselves are Villarceau circles. Banchoff (1990) has explored such a torus with computer graphics imagery. One of the unusual facts about the circles is that each links through all the others, not just in its own torus but in the collection filling all of space; Berger (1987) has a discussion and drawing.
Read more about this topic: Villarceau Circles
Famous quotes containing the words filling and/or space:
“Science has done great things for us; it has also pushed us hopelessly back. For, not content with filling its own place, it has tried to supersede everything else. It has challenged the super-eminence of religion; it has turned all philosophy out of doors except that which clings to its skirts; it has thrown contempt on all learning that does not depend on it; and it has bribed the skeptics by giving us immense material comforts.”
—Katharine Fullerton Gerould (1879–1944)
“In the tale proper—where there is no space for development of character or for great profusion and variety of incident—mere construction is, of course, far more imperatively demanded than in the novel.”
—Edgar Allan Poe (1809–1849)