Such lines are called Ammann Bars. This is one of Ammann's tilings with bars shown in color. The tile edges also align to form straight lines, but sooner or later there is some interruption to the pattern. Nevertheless, the Ammann bars continue across the entire tiling. Hierarchy Most aperiodic tilings have some kind of heirarchical structure.
Composition is one variety.
This Ammann Tiling shows another. The asymmetrical notches force tiles to align in certain directions as shown by the dark purple lines. These lines generate a hierarchy of progressively larger H-shaped patterns. Just about everything there was to know on the subject at the time.