© | Dror Bar-Natan's Image Gallery: Knotted Objects:

Planet Hopf

Hopf says that 3-space fibers over the 2-sphere, the Greeks say the 2-sphere is planet Earth, and Copernicus says it spins.
The consequences are ridiculous.

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Alternative version

Image/Video Downloads: PlanetHopf2.png (2400x2400), PlanetHopf2_720.png (720x720), PlanetHopf2.ogg (720x720), PlanetHopf2_240.ogg (240x240). Source files: Academic Pensieve: Projects: PlanetHopf.

This image is best viewed at full size (2400x2400) and at full screen (hit F11)

What's shown? The pullback to S3 of a (partial) map of (spinning) planet Earth, via the Hopf fibration S3→S2, projected down to R3 by the stereographic projection. Much like the cheese wheel on the left, a wedge-shape of openning 80o is cut out of R3. The whole map of Earth can be seen on both sides of the cut (each in itself, a half-plane), with an 80o rotation between the sides. For improved visibility, the full pullbacks of only Canada, China, Italy, Colombia, and South Africa are shown, and even these are made almost entirely translucent as they pass through the cut wedge.
A home made cheese wheel by David Fankhauser.