Den fredagen den 2:e augusti 2013 kl. 08:44:38 UTC+2 skrev David Bernier: > On 08/01/2013 08:25 PM, Richard Tobin wrote: > > > In article <firstname.lastname@example.org>, > > > <email@example.com> wrote: > > > > > >> And i may claim that Pi is a rational, because i know i can write it as > > >> a continued fraction. > > > > > > But that isn't what rational means. > > > > > > -- Richard > > > > > > > This made me think of what happens if we try to apply the Euclidean > > Algorithm to sticks of length (1+sqrt(5))/2 and 1. > > > > Of course, (1+sqrt(5))/2 = phi = 1.618.... . > > Then, phi -1 = 0.618.... = 1/phi. > > > > So, after chopping off 1 unit from the stick of length phi, > > it now has length 1/phi, and the two sticks have lengths in > > ratios 1 :: 1/phi or phi :: 1. > > > > phi = 1+ 1/(1+1/1+1/(1+ (to suggest the continued fraction expansion). > > > > Yet, phi is irrational. > > > > david >
But one 1/phi is not a rational ratio, but the 1/6 hexagon (center to vertex/sum of sides) is and so is all polygons derived from multiplying the vertices.