Dual of a pattern
As shown in Line Art (2), the repetition pattern is sometimes as much interesting as the original pattern – or even more interesting. Since it's also a pattern and closely related to the original one, it's called the "dual". The background pattern of these webpages is a good example. In itself, it's quite a boring pattern, so let's see what the dual looks like. It starts with four diamonds (1a). The tops of the diamonds are connected two by two (1b), and from adjacent angular points of the diamonds lines are added, parallel to the connecting lines (1c). Everything fits neatly on a rectangular grid (checkered paper!).
Connected diamonds
1. Connected diamonds
This pattern is repeated by copying and mirroring (2). (This in itself gives rise to a new dual!). Only some "openings" remain.
Pattern repeated
2. Pattern repeated
When the openings are filled, the space is covered up in a nice way (3).
Space covering pattern
3. Space covering pattern
When all intersections are taken as a center for a small square (size of a check of the checkered paper), it becomes clear, that this is the dual of the background pattern (4). (Grid of the dual and background pattern are shifted relative to each other!).
Background pattern's dual
4. Background pattern's dual
Here it is.
Background pattern
5. Background pattern

Let's go back to the dual pattern. Remarkably enough, it can be viewed in a completely different way. It then consists of two copies of an odd-shaped pattern, one horizontally, one vertically oriented (6), and repeated over a diagonal grid (in red). No diamonds at all!
Repetition base
6. Repetition base and grid
The interesting thing is, that the horizontal and vertical patterns can be copied-and-shifted independently. If done so, the horizontal one in a horizontal, and the vertical one in a vertical direction, both halfway their own pitch, then this comes out (7). Looks like cats! Notice, that "circles" appear (again, see also 6d in Line Art (1)), although the whole pattern only consists of straight lines.
7. "Cats"
A completely different picture appears, if both vertical patterns of the previous step are shifted diagonally. The "circles" are gone and replaced by squares and eight-pointed stars (8).
Background's Dual
8. Squares and eight-pointed stars
Finally, this complete pattern can be copied-and-shifted diagonally, and an attractive final pattern comes out (9).
Background's Dual
9. Final pattern
So, sometimes, the dual of a (boring) pattern - and its many variations - can be much more appealing than the pattern itself!

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All drawings and figures, copyright © 1992, 2009, Zef Damen, The Netherlands.
Personal use only, commercial use prohibited.

Nederlandse versie Nederlandse versie Last updated: 16-November-2009