Saturday, June 27, 2009

New Sequence

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.

Friday, June 5, 2009

Hackers and Painters

Hackers and Painters, the essay:

Hackers and Painters, the book:

Gratuitous tile set image made with a Flash Air program written by a painter: