I finally decided to include
continued fractions in my project on
rectangles and spirals. While doing the research I read that "numbers with
periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients." My previous
rectangles and spirals were all based on irrational solutions of quadratic equations. So, this got me thinking that all I needed to do was start with a periodic continued fraction, and I might find another series of rectangle and spirals similar to the golden rectangle. The continued fractions for my second sequence are:
[1; 3,1,1,3,1,1,3,1,1, …]
[1; 7,1,1,7,1,1,7,1,1, …]
[1; 15,1,1,15,1,1,15,1,1, …]
[1; 31,1,1,31,1,1,31,1,1, …]
etc. . . .
So, I decided to start with a similar sequence. The continued fractions I came up with are:
[1; 2,1,1,2,1,1,2,1,1, …]
[1; 4,1,1,4,1,1,4,1,1, …]
[1; 6,1,1,6,1,1,6,1,1, …]
[1; 8,1,1,8,1,1,8,1,1, …]
etc. . . .
And, the corresponding ratios look like:
b/(a–2*b/3) = a/b
b/(a–2*b/5) = a/b
b/(a–2*b/7) = a/b
b/(a–2*b/9) = a/b
I haven't drawn the spirals, but I think they look a lot like my second sequence.
