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:
“I am filling the room
with the words from my pen.
Words leak out of it like a miscarriage.
I am zinging words out into the air
and they come back like squash balls.
Yet there is silence.”
—Anne Sexton (1928–1974)
“True spoiling is nothing to do with what a child owns or with amount of attention he gets. he can have the major part of your income, living space and attention and not be spoiled, or he can have very little and be spoiled. It is not what he gets that is at issue. It is how and why he gets it. Spoiling is to do with the family balance of power.”
—Penelope Leach (20th century)