What are the parameters for the edge-match-restricting arcs drawn on Penrose tilings?

Question

I would like to try generating some computer art involving Penrose tilings. I'm looking into the layouts algorithms separately, this question concerns the decoration.

Here is a schematic of P2:

No problem drawing the boundaries, but I also understand that we have to add rules as to which edges can go where. There are a few ways to represent this but the one I am interested in is this:

I assume these are arcs of circles whose centres are at the obvious points. Given the dimensions of the first figure, what are the radii of the blue and green circles on the left and right?

As an intuitive guess I would say $r_{blue}:r_{green}=1:\phi$ in each figure, but I would like to see how to arrive at this properly.

Also, it seems that if I increase $r_{blue}$ for the kite and decrease it for the dart that the blue arcs could be made to join for any chosen first radius, but if I did that (and changed the green radii to maintain the touch at the centre) would the green arcs still meet on the edges? If so, how would I generate the $r_{green}$s given the $r_{blue}$s?

Answer 1

The specific radii are not important. What is important is that:

• the radius of the green arc in the yellow figure and the radius of the green arc in the purple figure should add up to 1 (and be different from each other);
• the radius of the blue arc in the yellow figure and the radius of the blue arc in the purple figure should add up to $\frac{1}{\phi}$ (and be different from each other).

The point is to ensure that trivial tilings (e.g. using the parallelogram from your first picture) are ruled out.