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| Maths intro HyperView intro |
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Mathematical Survey |
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The purpose of HyperView is to visualize the 4-dimensional unit sphere
(from now on simply referred to as "the hypersphere"). Since the
hypersphere is a 4-dimensional object, we cannot see it directly and
first have to transfer it into 3D space in some way or another - our
method of choice will be stereographic
projection. Nevertheless there are some difficulties we have to deal with. The first rather obvious one is the fact that since stereographic projection (which maps the hypersphere into R3) is bijective, by naively projecting the whole hypersphere we would obtain the entire 3-space as the image which cannot quite be called a proper visualization. Thus we may only visualize parts of the hypersphere at the same time if we like to obtain nice images. The next question that naturally arises is which parts of the hypersphere (we will call them "hypersubsets") should be chosen for the resulting pictures to be as asthetic and informative as possible. An article in "Spektrum der Wissenschaft" (the german edition of Scientific American) dating back to 1988 and treating a visualization problem of a similar kind (namely, projecting the hypercube) in a (from my point of view) more than simplisitic way makes the suggestion to take a 4-dimensional equivalent to circles of latitude. This article being the main source of information for my first considerations in the subject, it confused me a lot because I didn't see any way to end up with the sort of pictures showing strangely distorted tori connected to each other and foliated in a peculiar manner. Only after extensive research throughout the web and a lot of headscratching I discovered information confirming my conjecture that circles of latitude are not at all sufficient and much more sophisticated hypersubsets have to be chosen. One way to do this involves the Hopf map, a continuous surjective map from the hypersphere to the common 3-dimensional sphere, and this is the approach taken in the program HyperView (details in the maths section). |
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© 2003 by Maximilian Albert • Anhalter42@gmx.de