Rectangular grid | Isometric grid |
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Fig.1 | Fig.2 |
Small grids are easily drawn by hand. Larger ones are best generated by computer on a laser printer. If you want a page-sized sample, click on the Fig.2 image which can then be printed.
first move | available directions | direction
1, repeated |
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Fig.3 | Fig.4 | Fig.5 |
The directions are numbered (see Fig.4). Zero is always straight ahead. There is no number three since that path is already taken.
For example, take path zero (straight ahead). Rules, once defined, are in effect for the duration of the pattern that is emerging. It’s easy to see that the straight ahead rule will lead to an infinitely long line to the right as it is reapplied at each subsequent grid point. Clearly, this rule at this time will not result in an interesting pattern.
If we choose direction one (a 60° right turn) then we will create a path that curves right, a hexagon, until it ends up at the starting point (see Fig.5). The arrow shows where we are currently. Here is a new situation we haven’t yet encountered. From our point of view, one path is already taken (direction 1). We can only go straight (zero), or two, four, or five. Remember the numbering is relative to the orientation of last segment. Zero is always straight ahead. We must establish a new rule which accounts for the cases where path one is taken.
This process is repeated – existing rules used wherever applicable and new rules created as required.
The smallest pattern has nine units: It starts and ends at the same point. Be prepared to see patterns exceeding many thousands of units.
What this site offers is a description of WORMS sufficient for you to write your own program. You will be able to delight in the discovery of many patterns. Many are beautiful not only in their finished state, but in their intricate growth behaviour along the way. You may even be able to answer the question “What is the largest pattern?”
If you’re just curious or have no desire to explore WORMS on your own, the remainder of this site is dedicated to the results of intense study by this author. All possible patterns are catalogued and illustrated. Various recurring textures are shown. Finally, some possible algorithms are reviewed.