3-D continued fractions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T13:36:34Z http://mathoverflow.net/feeds/question/111791 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111791/3-d-continued-fractions 3-D continued fractions Boris Frolov 2012-11-08T08:18:38Z 2012-11-08T08:41:48Z <p>Which theorems from classical theory of continued fractions have 3-(or multi-) dimesional analogs?</p> <p>Of cause classical one is a periodicity of Klein polyhedra. Probably there are some more...</p> http://mathoverflow.net/questions/111791/3-d-continued-fractions/111793#111793 Answer by Alexey Ustinov for 3-D continued fractions Alexey Ustinov 2012-11-08T08:24:52Z 2012-11-08T08:41:48Z <p>(1) There is 3-D analog of Vahlen's theorem, see <a href="http://link.springer.com/article/10.1007%2Fs11006-006-0018-6?LI=true" rel="nofollow">http://link.springer.com/article/10.1007%2Fs11006-006-0018-6?LI=true</a></p> <p>(2) 3-D isolation theorems and extemal Davenport forms (see Cassels "An Introduction to the Geometry of Numbers" and Swinnerton-Dyer, "On the product of three homogeneous linear forms" Acta Arith., 1971, 18, 371-385). The key role here play numbers $2\cos\frac{2\pi}7$, $2\cos\frac{4\pi}7$, $2\cos\frac{6\pi}7$ (3-D Golden Ratios).</p>