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.

Hackers and Painters

Hackers and Painters, the essay: http://www.paulgraham.com/hp.html

Hackers and Painters, the book: http://www.paulgraham.com/hackpaint.html

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

Radical Non-periodic Tiling

This tiling is from a set of seven tiles, including the pentagon. It's the latest in a series of tilings.

Animated Radial Tiling

Here's an initial test version of animated radial dart-rhombus tilings.

Tile Sets

I have my Air tiling program under control, and I'm using it to generate asymmetrical tilings from tile sets I design myself. The tile sets are all periodic, could be used to generate a variety of symmetrical tilings, but I arrange them in asymmetrical and non-periodic tilings.

Tiling No. 2

Possibly out of control. From the same tile set as No. 1.

Tiling No. 1

This is the first successful test of a new tiling program. The program allows me to create custom tile sets, and tilings from the tile set. This is still just a preliminary test of the program, to show that the code generally works, so the colors are not important

This particular tiling is one I tried to draw by hand. The tiling could be extended to fill the plane. It's non-periodic, but the tile set itself is not aperiodic. This is a great improvement on my previous tiling program, which generated only radially symmetric, dart-rhombus tilings. Having the ability to do asymmetric tilings is liberating.

This is my first program with AS3, my first Air program, and my first OOP program. I'm excited.