The two kinds of kites of a kite pair are always arranged in rings. This way they are enclosing an inner or secondary grid. The inner grid is tilted by the angle λ of the kite against the primary grid. The rings consist of a number of kites of one chosen kite pair, but all of them must have the same orientation. The red line marks the dragon line separating the grids with different orientation. This fundamental property is crucial for solving puzzles with kite tiles.
The ring orientation in the following example is counter clockwise. The kites are of type 2-1, i.e. the base triangle of the main kite has legs of 2 and 1. The corresponding complementary kite has legs of 3 and 1 unit length. Given the inner grid shape, there is no other way to enclose it with this kind of kites. The tile angle of the inner grid is λ = 2*arctan(½) = arctan(4/3) = 53°.1301…, which is a irrational number. If the construction is mirrored, the orientation of the kites is clockwise and the tilt angle negative.
Additionally we can modify the kites, if we allow diagonal cutting of two squares in the inner as well as in the outer grid. The following diagram shows some of the resulting kite variants. The border of the ring of kites is now no longer made of square units but of 2-1 triangles. Nevertheless the basic structure of the kite rings remains the same. We will find these ring forms in the Extended DiDom Puzzle, where the six possible kite variants create an interesting complexity.
Now we can repeat the ring construction inside the inner grid. Depending on the orientation of the second ring a third grid with tilt 2*λ or 0 is enclosed. This process con go on recursively, als long there is enough space. All grids will be tilted by the angle n*λ, where n is a positive or negative integer, if only kites of the same pair had been used. Because λ is irrational the resulting angles are all different, so the grids never match besides they are separated anyway.
More complex structures appear, if the inner rings are composed from rings of other kite pairs. On top of that the inner grid may harbour many equal or different ring types. From the view point of the outer grids all look the same, a big block of squares.
Kite pairs of the triangle grid form rings in a similar way. They enclose an inner triangle grid, which is tilted by the angle of the main kite of 2*arctan(1/5*√3) = arctan(5/11*√3) = 38°.2132… against the base grid.
