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:
“the gap of today filling itself
as emptiness is distributed
in the idea of what time it is
when that time is already past”
—John Ashbery (b. 1927)
“The true gardener then brushes over the ground with slow and gentle hand, to liberate a space for breath round some favourite; but he is not thinking about destruction except incidentally. It is only the amateur like myself who becomes obsessed and rejoices with a sadistic pleasure in weeds that are big and bad enough to pull, and at last, almost forgetting the flowers altogether, turns into a Reformer.”
—Freya Stark (1893–1993)