You've probably seen images of the Hopf fibration, a nice decomposition of 3-dimensional space (or more accurately the 3-sphere) into nested tori, and of the tori into circles, so that every point of the space belongs to one of the circles (including the z axis, which can be interpreted as a circle through the single point at infinity) and every two circles are linked.
Here's a very similar construction: a decomposition of 3-dimensional space (or more accurately projective 3-space) into nested hyperboloids, and each hyperboloid into lines, so that every point of the space belongs to one of the lines (including the line at horizontal infinity) and every two lines are skew.

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Here's a very similar construction: a decomposition of 3-dimensional space (or more accurately projective 3-space) into nested hyperboloids, and each hyperboloid into lines, so that every point of the space belongs to one of the lines (including the line at horizontal infinity) and every two lines are skew.

( Read more... )
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