Periodic, isotropic weaves from overlays of three quasi-cartesian grids

Three copies of the cartesian grid cannot be overlaid into a pattern that is both periodic and isotropic. This is a familiar problem in designing halftone patterns for printing. The traditional arrangement of three square-grid halftones avoids distracting moire patterns with a 120° rotation dispersal that results in a pattern of dot 'rosettes' that is isotropic but not periodic. Wang and Loce show the way towards periodicity. If the orthogonal grids are not square but 1:0.866 rectangles (i.e, based on the base-to-altitude ratio of equilateral triangles), and still arranged 120° to each other,  isotropic and periodic patterns are possible.

Three copies of a rectangular grid with aspect ratio 0.866 can be rotated 120° to each other and still form periodic patterns.

We may prefer, instead of sacrificing the equilateral property of the cartesian grid, to sacrifice its orthogonality. Three copies of a slightly sheared (skew) cartesian plane can be overlaid with 120° rotation dispersion to form periodic patterns. (The sheared grid is formed by rotating two copies of a parallel ruling ± 43.898 degrees.)

Three copies of a slightly sheared cartesian grid can be rotated 120° to each other and form periodic patterns.

A cartesian grid has two line directions, after three copies are dispersed 120° from each other, there are six line directions (see figure below.) It is somehow easier to perceive these six directions as three narrow-angle 'flashlight beams' with (half-angle) beam-spread of 15°.

Three right-angles rotated 120° to each other tend to be perceived as three narrow beams with half-angle 15°.

In Panda, Maulik, Chakraborty, and Khastgir there a several periodic solutions for billiards played on an equilateral triangular table. Any of these solutions that preserve the full symmetry of the triangle could decorate the equilateral triangles of a deltahedron resulting in a surface wrapped by geodesic lines.

Periodic solutions to billiards on an equilateral triangle table (from Panda, Maulik, Chakraborty, and Khastgir.)

The solution labelled (m=2, n=5) caught my eye, me thinking that the lines were orthogonal (in fact, they are just slightly oblique.)

Flat weaving in this pattern would look something like this:

Flat weaving the (m=2, n=5) periodic solution to billiards-on-a-triangle. Heavy lines emphasize the kagome-like organization.

Extended in this way, it becomes easier to see that the lines are not quite orthogonal. This geometric construction shows that an angle that needs to be 15° for orthogonality is actually atan(sqrt(3)/7) = 13.8979°.

For lines in the periodic pattern to cross at 90°, the apex angle of the green triangle would need to be 15°. This geometric construction shows that it is instead arctangent of the ratio of one equilateral altitude to 3.5 equilateral sides.