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.

The post Wheel-Within-a-Wheel Exploration with Wild Gears appeared first on SpiroGraphicArt.

Pattern made with the four wheels that make 7-points with ring 150/105.

There are four wheels that will give you a 7-pointed figure when used by the 150/105 ring: the 30, 45, 60 and 75.

Here they are combined in one design, using the new Spirograph Deluxe Set and the red, green and blue pens that came with it.

How to make it

I started with the smallest 30-tooth wheel, using holes 1, 2 and 3 with the green pen, to make the outermost set of patterns. Remember that you have to line the holes up carefully with a mark on the wheel before beginning to draw each pattern.

Then I took the 45 wheel, skipped the #4 hole and drew with holes 5, 6 and 7 with blue.

With the 60 wheel, I jumped ahead to 9, 10 and 11 using the red pen.

Finally I used wheel 75 to draw the final set using holes 13, 14 and 15 in green again.

Try it!

The math part

Notice how these four wheels, 30, 45, 60 and 75, are all multiples of 15!

If there were a tiny 15-tooth wheel, it would form a heptagon, without any crossing lines, because the inside of the 150/105 ring has 105 teeth, and 105 divided by 15 is 7. So the 15-toothed wheel would go around 7 times inside the 105-toothed ring before it arrived at the starting point.

Wheel 30 crosses one point place on its way to the next point, and goes around twice to meet the starting point.

Wheel 45 crosses two points and goes around 3 times to get back to the starting point.

Wheel 60 crosses 3 points, and wheel 75 crosses 4 points.

Trace these patterns with your finger or mouse in the design above and you’ll see it.

The post Seven-Pointed Spirograph Designs appeared first on SpiroGraphicArt.