A bit of a windy road: starting, as usual, with the personal frame of reference....
PyGeo's current implementation supports the exploration of the geometry of complex numbers, and therefore speaks Mobius transformations. http://pygeo.sourceforge.net now has a pretty picture of a simple recursive transformation of 4 circles on the unit sphere (...thanks to __iter__ the ability to recursively transform any arbitrary set of geometric objects is now built-in to PyGeo). My current exploration (current as in today) is finding the mechanism to build a Mobius transformation that would be based (in part) on it's (pickable and movable) fixed points - of which a Mobius transformation has 2, which may coincide, or be located inconveniently - e.g. at infinity. Which has me stepping into the math of the fixed points of a function - it being trivial to find the fixed points, given the Mobius transformation matrix, but less trivial (from where I am sitting at the moment) to build the transformation from fixed point information. So I am struggling and researching some. In the course of which I come across this definition of "Fixed Point" in a programming glossary, @ http://carbon.cudenver.edu/~hgreenbe/glossary/second.php?page=F.html Of a function, f:X-->X, f(x)=x. Of a point-to-set map, F:X-->2^X, x is in F(x). The study of fixed points has been at the foundation of algorithms <http://carbon.cudenver.edu/%7Ehgreenbe/glossary/second.php?page=A.html#Algorithm>. Having discussed here my growing interest in some study of algorithmics, but not getting there yet, but pursuing something that as far as I am aware is unconnected to such study, and then finding this statement indicating there is more of a connection - perhaps - than I had understood, is to me interesting. I have thought of "fixed point" (in programming) as connected to/opposed to "floating point", not as something directly connected to the concept of "f(x)=x" The statement above seems to be telling me otherwise. Guess I am fishing for some exposition on the statement that the "The study of fixed points has been at the foundation of algorithms" Art _______________________________________________ Edu-sig mailing list [email protected] http://mail.python.org/mailman/listinfo/edu-sig
