In March 2022 I was contacted by a person named Glagolj who has mathematically described what I’ve been calling “wheel-within-a-wheel” designs made with Wild Gears.

He prefers to call it “second order roulette” as it involves rolling one curve on top of another curve. Fair enough. For me, a mystery. For Glagolj, a mystery to be figured out and encapsulated in a parametric equation.

He wanted to use the patterns I’d drawn by hand and compare them side by side with the renderings generated by his javascript plotter. That’s the ultimate check on the formula. As you can see here, it works.

The plotter can be found here. It allows you to plug in the four gear numbers of a second order roulette, and generate the pattern they should give.

Glagolj’s article on the formula is here on Github.

He prepared a couple of slideshows with some of the interesting patterns he found. These are all computer-generated, so he can use any gear he wants. Of course, working with physical gears we have to work with the ones we have. But the numbers are in the upper-left if you want to try drawing them yourself.

Here is the second slideshow by Glagolj:

Several people have suggested I try third-order roulette patterns, i.e. a wheel-within-a-wheel-within-a-wheel. Well, Glagolj is ahead of me there. Now I have to try to see if I can duplicate his computer-generated results.

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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.

I do know, however, that if your small gear isn’t much smaller than the hole in the larger gear you’re using, you get loops. This pattern has lots of loops, as it uses a #40 gear in a #45 hole. The larger wheel is 140 and the large ring is 180. All except the 40 are in Wild Gears’ Full Page Gear Set. The 40 came from the Plentiful Gear Set.

It’s fun to watch the pattern emerge, so I made a video of it (below). I messed it up 5 times before I gave up and used the blooper anyway. I tried again without the camera running and got it perfect the first time. Go figure.

The first time I discovered this combination of gears, I found the pattern interesting and coloured it. So I made it in black as a colouring page, and here are some results. A couple are done by an Australian correspondant, Suze.

In case you’d like to try colouring it yourself, you can buy the colouring page as an instant printable. Send me your pictures and I’ll add them to the gallery.

In the video, I’m using a Micron 05 pen (.45mm). This is a pricier pen than most that I use, but I wanted some archival ink to create more serious artwork with my Wild Gears and Spirograph. Christmas is coming, after all.

Colouring page available here.

Shop Wild Gears at this link.

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How many points will you get? It depends on the same mathematical formula that determines the number of points of a hypotrochoid. You don’t have to do the math, though. Just use this Table of Spirograph Points in the reference section of this website.

Once you’ve drawn your epitrochoid, try drawing a design inside the ring to match it before you take the ring off the paper, as I did in the video.

Here’s another idea: use it as a picture frame or greeting card with a photo or message on the inside!

You can also draw an epitrochoid around another wheel, rather than a ring. Try it.

Drawing around a Spirograph rack, or around open shapes made with Super Spirograph parts, also creates epitrochoids, but that’s a subject for another video.

The post Video: Drawing on the outside of the ring appeared first on SpiroGraphicArt.

With ring 144/96, the bar produces a 12-pointed pattern. With ring 150/105 it makes 21 points. So using the Table of Spirograph Points, we see that it matches wheel 40. However, at it’s largest diameter it’s bigger than wheel 40.

If you’ve been avoiding the bar, I hope this video will encourage you to give it a try.

The post Intro to the Bar Wheel appeared first on SpiroGraphicArt.

How to know in advance how many points a given wheel will make in any Super Spirograph layout. 

Super Spirograph comes with the same rings and wheels as the regular Spirograph plus the “super” parts shown in the picture, which you snap together to make a big layout.

If you want to predict the number of points that a certain wheel will give you in a particular layout, you need to know the number of teeth in the layout.

You can then use the table on this page to predict the number of points the design will have.

Chart: Number of teeth on the different super parts

Note that the curves have more teeth on the convex side than the concave side. Some layouts have a combination of concave and convex curves on the inside.

Sample calculations

Rectangles:

A rectangle with a straight piece F on the shorter side and an F plus E on the long side, plus a curve C at each corner: would have:

4 x C (concave) plus 4 x F plus 2 x E = 4×24 + 4×56 + 2×20
(Remember your grade school math and multiply each group before you add!)
= 96 + 224 + 40 = 360 teeth on the inside

A smaller rectangle allows you to draw designs outside as well as inside on letter-sized paper. This one has one long F piece on each long side, and a short E piece on each short side.

Inside: 4 x C plus 2 x F plus 2 x E = 4×24 + 2×56 + 2×20 =
96 + 112 + 40 = 248 teeth on the inside

Outside: Note that only the curves are different lengths on the outside.
4 x C plus 2 x F plus 2 x E = 4×36 + 2×56 + 2×20 =
144 + 112 + 40 = 296 teeth on the outside

See some Super Spirograph designs drawn with these two rectangles.

Cloverleaf

The cloverleaf is a very curvaceous layout. It is formed from the curved pieces laid out D-B-C-D-B-C-D-B-C-D-B-C, with the C pieces turned in the other direction from the D-B units. To count the number of teeth, use the concave number for D and B, and the convex number for C.

4 x D (concave) plus 4 x B (concave) plus 4 x C (convex)
= 4×32 + 4×16 + 4×36 = 336 teeth on the inside.

See some Super Spirograph patterns made with the cloverleaf layout.

Curvy diamond

My old Super Spirograph booklet doesn’t even have a name for this layout, it just says “try this”. So I’m calling it the curvy diamond, and I like it a lot. It’s small enough that you can draw designs on the outside on letter-sized paper, as well as on the curvy inside.

The east and west wings are two C curves. The north and south wings are a single D curve. The four wings are joined together with B curves faced the other way.

Inside: 4 x C (concave) plus 2 x D (concave) plus 4 x B (convex) = 4×24 + 2×32 + 4×24 = 256 teeth on the inside.

Outside: 4 x C (convex) plus 2 x D (convex) plus 4 x B (concave) = 4×36 + 2×48 + 4×16 = 144 + 96 + 64 = 304 teeth on the outside.

I know the number of teeth. So now what?

Now go to the big chart and match your wheel numbers (rows) with your layout numbers (columns) and find the number of points your pattern will have.

The post Super Spirograph by the Numbers appeared first on SpiroGraphicArt.

TRIGGER WARNING: If you were severely traumatized by math in high school, you might want to avoid watching this video. However, if you are in therapy for Math Class PTSD, you may wish to try small doses of this video to benefit from the narrator’s calm, relaxing voice.

VIEWER WARNING: The more you know about math, the more you’ll appreciate this video. For example, if you have forgotten what a cosine is, it’s likely to go right over your head. In that case, perhaps you will sleep more easily knowing that someone has deconstructed the magic generated by all those toothed rings and wheels. Or not.

The post Deriving the mathematical formula for the pattern made by a Spirograph gear appeared first on SpiroGraphicArt.

You’ll find it in the menu under “Reference section”. It was under “math” but maybe that’s too off-putting and people won’t look there. The reference section also has tables showing the number of points made by any combination of teeth.

There are other nerdy, technical posts that I’m collecting under the same banner. Some people will go for the math and others will be repelled. I just want you to find it if you need it.

The post Spirograph Pattern Guide appeared first on SpiroGraphicArt.

144/96 wheel 45

Ever since I was a kid, I wanted to know:

How can you predict how many points (or “petals”) a spirograph pattern will have?

There IS a formula. You need to know the number of teeth on the wheel and the number of teeth on the ring.

The number of points (N) is equal to the Least Common Multiple (LCM) of  the number of teeth on the ring (R) and the number of teeth on the wheel (W), divided by the number of teeth on the wheel.

In a spreadsheet, you would use “= LCM(R,W)/W”. Replace R and W with the coordinates of the respective cells.

In the case of the pattern above, the least common multiple of 96 and 45 is 1440. One way to find this is by prime factorization. Find the prime numbers which

For example, for LCM(96,45) we find:

Prime factorization of 96 = 2 * 2 * 2 * 2 * 2 * 3 = 2⁵ * 3¹
Prime factorization of 45 = 3 * 3 * 5 = 3² * 5¹
Using the set of prime numbers from each set with the highest exponent value we take 2⁵ * 3² * 5¹ = 1440, which is the LCM.
Divide the LCM by the wheel, so 1440/45 = 32, the number of points in the design above.

Find a table of Spirograph Point Numbers here.

The post Calculating the Number of Points in a Spirograph Pattern appeared first on SpiroGraphicArt.

Ring 144/96, wheel 72

A reader asked whether the hole numbers on the Spirograph wheels correspond to the distance from the edge in millimeters. The answer is “not quite”. They are very regular, however, which is one of the reasons that the designs are so interesting and appealing to the eye.

So how to measure how far apart the holes are from each other in terms of the radius of the wheel?

It’s hard to tell by looking at the wheels, because the holes are laid out in a spiral. That’s because the diameter of the holes is bigger than the radial distance between them. If the holes were laid out in a line, they would overlap. The integrity of the round holes would be gone. You wouldn’t be able to draw patterns, and the wheel wouldn’t hold together.

However, when you draw a design like this one, you line up each hole at the same spot on the ring before you draw its individual pattern. Let’s look at the lines drawn by the holes in this pattern and measure the distance between them.

This pattern is drawn using holes 1, 3, 5, 7, 9, 11, 13, 15 and 17. It was drawn with my old Spirograph.

Since the distances are small and hard to measure individually, let’s measure a number of them together and divide.

From hole 1 to hole 17, with the best accuracy and precision as I can manage with my little vernier caliper, is 10.4 mm (0.409 in).

Divide 10.4 mm by 16, because there are 16 spaces between holes 1 and 17, and you get 0.650 mm (0.0256 in).

My result: 0.650 mm (0.0256 in).

This result should be independent of the size of the holes and the size of the pen used, as long as the holes are the same size and you use the same pen throughout.

I also think it’s the same with all the wheels, because that’s how the designers of the set made it. To my eye, all the patterns produced have a similar distance between the lines, but I haven’t measured this yet. I hadn’t actually asked myself the question until doing this, but I’ll experiment and report in a future post.

I used the wheel and ring from my old Super Spirograph set for this particular drawing. Comparing designs made with the old set and the new set shows that the old set is more precise. I’ll make more posts about that in the future. Quality really shows when you’re drawing fine lines close together.

Like any good scientific experiment involving measurements, others should try it themselves and see if they can replicate my results. Let me know how you do in the comments below.

(Hmmm…. my inner science teacher is showing!)

The post What is the radial distance between the holes in Spirograph wheels? appeared first on SpiroGraphicArt.