On Monday, August 11, 2003, at 07:25 AM, Jon Perry wrote:

Anyone out there can come up with a comment on the symmetric nature of sin, cos and tan?

sin(x)=sin(-x) Actually, it's cos(x) = cos(-x) --- the symmetry points of sin and cos are where it takes values 1 and -1. As a complex function, the symmetry of cos (or sin) is one of the 7 strip patterns, generated by two 180 degree rotations (about 0 and about pi) or alternately by one 180 degree rotation and one translation (cos(x) = cos(x+2pi). The quotient space of the plane by the symmetries of cos is topologically a plane, but geometrically it is an infinitely long pillowcase, with two finite corners, but open at the infinite end. (you can take the strip 0 < Re(z) < pi: this is a fundamental domain for the symmetry, and fold each of the two edges in half across the real axis). In orbifold notation, an efficient way to analyze and describe all two-dimensional symmetry groups, the quotient is the (2 2 infinity) orbifold. Cos maps the quotient orbifold homeomorphically to the complex plane.

Holomorphic functions are incredibly closely linked to topological descriptions, and in particular, any other function with the symmetries of cosine that is injective on the quotient orbifold has the form f(z) = a cos(z) + b, for constants a not 0 and b arbitrary. Bill Thurston

Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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