It's not entirely clear what you have in mind by the "dimension" of a field in this context. If you're looking for an algebraic framework in which to do Euclidean and related geometry in higher space, the closest it's possible to get to a field is a generalisation of complex numbers and quaternions called "Clifford algebra" --- see http://en.wikipedia.org/wiki/Clifford_algebra http://en.wikipedia.org/wiki/Geometric_algebra A Clifford algebra is a ring, but not in general a field: multiplication is non- commutative, and nonzero elements may be isotropic (singular; lacking reciprocal). But it is still possible to do some analysis (calculus) in this environment. The situation is analogous to matrices, which provide a more traditional (though much clunkier) approach to geometric computation. The geometric algebra specific to Euclidean 3-space is Cl(3,0,1) or DCQ's. Curiously little information is available about these --- a brief intro and toy application is posted at https://docs.google.com/leaf?id=0B6QR93hqu1AhNDdmOGRhM2QtZThjNy00ZTc4LWI4MzY... Most practitioners prefer to use the larger algebra Cl(4,1,0) or CGA, partly because it is slightly more mathematically respectable, but mostly on the grounds of incorporating the Poincare group for Minkowski spacetime (aka conformal or Moebius 3-space). There seems no obvious reason why the quaternionic version of fractals should not generalise further --- given sufficient algorithmic dedication and computational muscle! Fred Lunnon On 2/21/11, Mike Stay <metaweta@gmail.com> wrote:
On Sun, Feb 20, 2011 at 7:30 PM, David Makin <makinmagic@tiscali.co.uk> wrote:
OK enough about me, does *anyone* know of a 3D or higher mathematical "field" satisfying all the axions on Wolfram here:
Well, there's a long history of quaternionic fractals. http://www.google.com/images?q=quaternion+fractals
There are also finite fields of arbitrary degree, though I don't know how you'd talk about convergence in a finite field. You could color the points by their period under the update function, I suppose. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun