A stereographic projection of a Clifford torus performing a simple rotation through the xz plane. In geometric topology, the Clifford torus is the simplest and most symmetric Euclidean space embedding of the cartesian product of two circles S1a and S1b. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1a and S1b each exist in their own independent embedding spaces R2a and R2b, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y. Got that? đ #science #mathematics #geometry
<p>Jason Hise</p>
More from Science & Technology
Synchrotron Soleil General diagram of Synchrotron Soleil. The outer circular ring is the synchrotron, i.e. a particle accelerator that brings electrons (light blue beam) to very high speeds. The electrons are accelerated by electric fields in the straight sections between green squares. The red rectangles are magnets that bend the beam. When the beam is bent the electrons emit synchrotron radiation (shown in yellow), especially X-rays; these are sent into the various beamlines (the straight lines branching out of the synchrotron). Each beamline contains scientific instruments, experiments etc. and receives an intense beam of radiation. #science #technology #physics #astrophysics
More from Science & Technology
Synchrotron Soleil General diagram of Synchrotron Soleil. The outer circular ring is the synchrotron, i.e. a particle accelerator that brings electrons (light blue beam) to very high speeds. The electrons are accelerated by electric fields in the straight sections between green squares. The red rectangles are magnets that bend the beam. When the beam is bent the electrons emit synchrotron radiation (shown in yellow), especially X-rays; these are sent into the various beamlines (the straight lines branching out of the synchrotron). Each beamline contains scientific instruments, experiments etc. and receives an intense beam of radiation. #science #technology #physics #astrophysics
