Compound Patterns, Part 1

I want to talk today about patterns, specifically about some of the oddities. For simple patterns, those created with a gear inside a ring, the formula for determining the number of loops that will result is well known (LCM(RingTeeth,GearTeeth)/GearTeeth) and with a little experience it's usually not difficult to have some idea of what the pattern will look like before you begin. There are usually not any big surprises with simple patterns. 



One of the differences between Wild Gears and Spirograph is the ability to do compound patterns with a gear inside a ring inside a gear inside a ring. 

Configurations like this can lead to highly unpredictable and incredibly complex patterns. Sadly, to my knowledge there has not yet been discovered a formula that will accurately predict the number of loops in the pattern produced by these multiple gear configurations. I've worked on it some but my math skills are not up to the task. If you know the formula, by all means, please leave a comment.

The complexity of some of these combinations is partly what lead me to using pencil quite a lot. Some combinations produce patterns so intricate and complex you'd never be able to do them with pen as the ink would saturate the paper and eventually lead to tearing. Examples of patterns like this are in the blog header graphic. 

The 120 Gear Set was designed mostly as a means of exploring these multiple gear patterns. Some of the most interesting and troublesome combinations are created with the 120 gear with inset rings of 24 through 29 teeth. The 29 ring in particular creates complex patterns that can be difficult to complete with pen. Even with pencil when you start getting into patterns that have 1,000 or more loops it becomes too much. But some of the combinations that are right near the edge of being too much produce patterns that are really quite beautiful. 

This is one of the patterns from the header background. It's 176/120/29/21/1 and was done with 0.5mm F lead. I don't think the center of this drawing could have survived being done with ink, certainly not with the pens I typically use.

The fact of 29 being a prime number means most of the gears you can use in that 29 ring produce patterns with a lot of loops. Combined with the tendency of the 120 gear in the 176 ring to produce patterns with 22 primary lobes, most all the patterns using the 29 ring in the 120 gear are complex. 

This is another of the patterns from the blog header background. This one is 176/120/29/23/1. It looks a tad more precise because I used 0.3mm 2H lead. The pattern is slightly different than the previous one but is clearly very similar. 

Here's the surprise.

This is 176/120/29/22/1. The small gear is just one tooth off from each of the previous two patterns, but the resulting pattern is radically different. I have no explanation for why this combination produces a pattern of only 22 loops when one tooth more or less on the inner gear produces vastly more loops. (This pattern also is in the header image. It is actually the same size as the other two. I had to shrink it to fit in the available space in the banner. I played with the colors on the scan of the drawing as the original was difficult to see. The lines aren't this dark on the original.)

All the other gear combinations in that 29 ring produce patterns much more like the first two above and nothing at all like the one just above. 

If you're wondering what a pattern of 1,000+ loops would look like...

This is 176/120/96/70/B1. It was done with 0.3mm 3H lead. The image has been tweaked to make it show up better as the original scan was quite light and difficult to see. There are 1056 loops in this pattern. As for the aesthetic quality... Hey, they're not all masterpieces. 

It is possible to take this compound thing to an almost insane level. With the sets I have I'm fairly certain I could do gear in gear in ring in ring in ring in ring. If you have the Enormous gear set I'm sure you could get a couple more rings in there before running out of room. 

Here is gear in gear in gear in ring.

This is 176/120/96/72/36/23/2. I gave up long before the pattern completed. While I didn't count the loops I'm fairly certain there are over 2,000 loops in the pattern as drawn and at least three times that many before it would complete. I gave up as the major pattern began to repeat for the third time. 

I'm sure there are some interesting and not entirely overwhelming patterns among the third and beyond levels of compounding, but without a generalized formula for predicting the number of loops in these patterns it seems not worth the trouble of investigating. One would need great patience with abandoning many drawings before finding one worthy of completion. 

Have you found interesting compound patterns? Let us know in the comments.