On second thoughts, this is just what's more usually referred to as "multidimensional continued fraction" --- googling which turns up a great raft of references, some of them remarkably recent. For example Multidimensional continued fraction and rational approximation Dai, Zongduo; Wang, Kunpeng; Ye, Dingfeng arXiv:math/0401141v1 [math.NT] [free download] Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions Pierre Arnoux, Valerie Berthe, Hiromi Ei, Shunji Ito Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 059–078 [free download as dmAA0104.pdf] Generation and recognition of digital planes using multi-dimensional continued fractions Thomas Fernique (2008) Multidimensional continued fractions Fritz Schweiger Oxford University Press, 2000 ISBN 0198506864, 9780198506867 234 pages [book] Surely this topic must have been discussed here before --- RWG in particular must have something to say about it! Fred Lunnon On 9/11/09, Mike Stay <metaweta@gmail.com> wrote:
I'm thinking about the problem of designing gear trains for a lego orrery which I'd like to be fairly accurate. The standard gear sizes are 8, 16, 24, 40, so I have ratios of 1:2:3:5 to work with. I figure that since lg(3) (where lg is log base 2) is irrational, you can get arbitrary accuracy out of a gear train. I think I need a + b lg(3) + c lg(5) ~= lg(period), with a, b, c small. Is LLL the right tool for solving this kind of thing?
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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