To understand more about the wheel-within-a-wheel phenomenon, I did a Wild Gears experiment. Video is below.

Inside a 120-tooth ring, I used a gear with 96 teeth, which has an off-center hole with 40 teeth. (I don’t remember which set this came from.)

Then I drew the pattern six times with six different small gears inside the 40 ring to see the patterns.

Looking at the math of it

Each time I got a 5-lobed pattern. That’s the basic geometry of the 120-96 gear combination, 5 loops or points, as you can see in the table here.

But there’s a lot of variation in how that 5-lobed pattern is expressed, according to which small gear is used.

Small gears used are: 32 (navy blue ink), 26 (pink). 34 (purple), 18 (green), 30 (red), 16 (turquoise).

Using the same table, looking at the number of points each gear would give in a 40-tooth ring, we see that:

• Gears 18, 26 and 34 all give 20 points
• Gears 32 and 16 give 5 points
• Gear 30 gives 4 points.

Now the differences in pattern density make sense. The three 20 point patterns (pink, purple and green) look different depending on the physical size of the gear.

The other three looser patterns make more sense if you try to count the points or loops within each lobe. Some of the points are hidden in the inner part of the pattern, so they’re hard to make out. But they’re there. You may get an idea from watching the video.

So the overall geometry of a wheel-within-a-wheel pattern depends on the larger gear/ring pair, and the density depends on the inner, smaller gear/ring pair.

Here’s the video, filmed on a sunny summer day in direct sunlight. The glass-topped patio table gives a nice flat surface for this work.