Hopf fibration

The three sphere is the set x2+y2+z2+w2=1 in R4. It is known under many different names. It appears in many different worlds: it is SU(2), the gauge group of the weak force. It is the set of unit quaternions. It is a two fold cover from all the rotations. It is a S1 "fibre bundle" over the 2-sphere (attach a circle at each point of a sphere). How can one see this object which lives in four dimensional space? One can look at the sphere, by foliating it by tori linked as brothers in arms and then turn things around. There is a general trick to see four dimensional space and that is stereographic projection. It is like when projecting the two dimensional sphere onto the plane. But in 3-space, we can now look at it. The space is now filled with nested tori made of circles (which we hopefully will detect in class in the 5th unit when cutting tori). The following visualization was designed by Thomas Banchoff (from Brown university who made a movie about it in 1985 (the same time than "Brothers in arms" came out but that is an accident). Tom then wrote a book "Beyond the third dimension" in 1990 where the 3 sphere is featured on the cover. You see a talk of Tom here talking about flatland and the fourth dimension). The animation below uses Povray code written by the math artist Paul Nylander in 2008. The movie was made for a differential geometry 230a class (with topic fibre bundles) in 2009 (substituting for Lydia Bieri who was at that time teaching the course). Heinz Hopf looked in 1931 at the map from the three sphere S3 to the two sphere S2: f(x,y,z,w) = [x2+y2-z2-w2,2(xw+yz),2(yw-xz) ]. Now, one can look at the places where f = c is a point on the 2-sphere which is a circle. The inverse image of a circle is a torus. One can now draw different tori, for example, by turning the circle around the 2-sphere and watch the tori move in the stereographic projection. This is what the movie shows. [ The music is "There are so many different worlds" from the movie "Bandits". It is the "Dire Straits" piece from the album "Brothers in Arms". Here is a live performance of Dire Straits playing the "many different worlds" in Wembley in 1985. The song is now a classic: See a Mark Knopfler performance from 2009, or a variation by Joan Baez (who began her career in Boston and Cambridge at Harvard square).]



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Here is a A javascript implementation of the Hopf fibration.