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 focused beam
folds all energy in:
the image glares filling all space:
the head falls and
hangs and cannot wake itself.”
—Archie Randolph Ammons (b. 1926)
“I would have broke mine eye-strings, cracked them, but
To look upon him, till the diminution
Of space had pointed him sharp as my needle;
Nay, followed him till he had melted from
The smallness of a gnat to air, and then
Have turned mine eye and wept.”
—William Shakespeare (1564–1616)