Border patterns … revisited

Border patterns, also known as freeze patterns, have been known to humanity for thousands of years, typically decorating the frame of a painting, mosaic, or some other two-dimensional design/pattern. They are straight symmetrical strips, of length finite by necessity but infinite in principle, and of constant, finite width. There exist seven types of them, discussed in several books, including my own ISOMETRICA (Chapter 2): p111, pm11, p1m1, p1a1, p112, pmm2, pma2. Below I am briefly discussing them from another point of view … with something else in mind (that may materialize in the future) 🙂
Border patterns are often considered to be one dimensional: this is quite imprecise, as there exist only two truly one-dimensional patterns, one with translation only (p) and one with reflection or rotation (m); below I show how these two patterns, appropriately placed at an upper and a lower level, can create six of the seven border patterns (the sad exception being pm11 — the only type besides p111 that has no level-swapping isometries).




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