From: Chris Hillman <hillman@math.washington.edu>
Newsgroups: sci.math
Subject: Re: Hyperfractions ? : approximants to sqrt(-1)
Date: Thu, 14 May 1998 18:32:09 -0700

On Thu, 14 May 1998, Peter Jack wrote:

> Hyperfractions ? : approximants of sqrt(-1)
> 
> I have a problem I'm working on. Maybe someone can help.
> 
> The basic idea is to find a representation of the
> hypercomplex numbers in terms of a sequence of
> rational numbers.

[... snip ...]

> Has anyone attempted such a construction before?

I don't know about the particular construction you outline, but you are
looking for a variety of "multidimensional continued fraction algorithm"
and there is an enormous literature on such things, mostly on continued
fractions in R^n but some concentrating on various hypercomplex numbers,
so it is quite possible someone has tried the approach you outline before.

Here are a few recent references which should give some idea of the
variety of approaches recently taken to this problem:

@article{djg:fn,
author ={David J. Grabiner},
title = {Farey Nets and Multidimensional Continued Fractions},
journal = {Monatshefte fur Mathematik},
volume = 114,
year = 1992,
pages = {35--60}}

@article{n:pcfa,
author = {A. Nogueira},
title = {The Three-Dimensional Poincare Continued Fraction Algorithm},
journal = {Israel Journal of Mathematics},
volume = 90,
year = 1995,
pages = {373--401}}

@book{s:fs,
author = {Fritz Schweiger},
title = {Ergodic Theory of Fibered Systems and Metric Number Theory},
publisher = {Clarendon Press},
address = {Oxford},
year = 1995}

@article{iko:jpa,
author = {S. Ito and M. Keane and M. Ohtsuki},
title = {Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm},
journal = {Ergodic Theory and Dynamical Systems},
year = 1993,
volume = 13,
pages = {319--334}}

@unpublished{l:skp,
author = {Giles Lachaud},
title = {Sails and {K}lein Polyhedra},
journal = {Contemporary Mathematics},
note = {to appear}}

@unpublished{l:kpgf,
author = {Giles Lachaud},
title = {{K}lein Polygons and Geometric Diagrams},
journal = {Contemporary Mathematics},
note = {to appear}}

@article{l:gmcf,
author = {J. C. Lagarias},
title = {Geodesic multidimensional continued fractions},
journal = {Proc. London Math. Soc.},
volume = 69,
year = 1994,
pages = {464--488}}

Hope this helps!

Chris Hillman
==============================================================================

From: edgar@math.ohio-state.edu (G. A. Edgar)
Newsgroups: sci.math.num-analysis,sci.math.research,sci.math,aus.mathematics
Subject: Re: Extending continued fractions
Date: Sun, 06 Sep 1998 09:21:22 -0400

In article <6srm44$678$1@news-1.news.gte.net>, aardwolf@gte.net wrote:

> 
> Now comes the $64 million question:  what objects comparable to
> continued fractions
> can be conjured from the situation for 3 dimensions?

A good question.  A 3-dimensional analogs of continued fractions
are studied in the branch of mathematics called "the geometry of numbers".
One source is H. Hancock, "The Development of the Minkowski Geometry
of Numbers" (2 volumes) re-printed by Dover.
-- 
Gerald A. Edgar                   edgar@math.ohio-state.edu