3D Printed Spirographs

Spirographs are a toy which originated in the 1800s. I wanted to prepare a fun, hands-on activity for a middle school math club as part of my job as Branham High School math club activities director and spirographs seemed like the perfect way to accomplish this. I used Fusion 360's parametric design feature to easily produce labeled spirograph wheels with different numbers of teeth. I then 3D printed a large number of these wheels and started spiraling! Additionally, I created my own original simulation using desmos.

• 3D printer
• Tape
• Blank paper or Index Cards
• Colored pens - I used these: https://www.target.com/p/pilot-12ct-g2-premium-retractable-gel-pens-fine-point-0-7mm/-/A-13317288#lnk=sametab
• Colored Pencils

I've attached two Fusion360 files: one for the inner wheel and one for the outer. I've also attached a large number of STLs with different wheel sizes. If you'd like to have other sizes start by downloading the two Fusion 360 files and follow the next step. Otherwise, skip to step 3.

First, click the modify drop down menu then the change parameters option. Change the parameter teethNum to the number of teeth you want on your gear. This will likely cause an error. When it does, right click the first extrusion in the timeline and select edit feature. Deselect all faces then select only the center face for the inner wheel or everything outside the the inner circle for the outer wheel. The label text will need to be adjusted manually since parameters can't be used in text. In the timeline right click the last sketch then select edit feature. Double click the text and change the text label to the same value as teethNum. If you choose a small enough gear number you may need to delete the circle in the center if it intersects with the other circles. The center circle is mostly unimportant since it will just draw a circle. It can be deleted by right clicking the second sketch in the timeline, selecting edit feature, then selecting the center circle and pressing delete. The model should then be ready for export. Right click the body in the bodies menu and select save as mesh.

The wheels will work in their original sizes, but I found they work best scaled by 150%. I was able to fit 4-5 parts on each print by nesting them inside each other. The print shouldn't require a raft unless your adhesion is poor. No supports are needed.

I created a simulation in Desmos - https://www.desmos.com/calculator/qbcxsb8old

In Desmos, change the variable s to the number of teeth in the inner gear and R to the number of teeth in the outer gear. Hitting play on c begins to draw the spiral. What's particularly great about this simulation is the ability to create spirals which aren't physically possible. For examples, making the variable l negative will move the "pen" to outside the the inner circle where it couldn't be physically drawn, but it can still be drawn on desmos: https://www.desmos.com/calculator/jjaqc8hbwc

You can also have "negative" wheel sizes: https://www.desmos.com/calculator/rprxpqvoq5

The math to make this graph isn't very relevant to actually drawing the spirals, but I think it will impress the middle school students!

Here's some math relevant to drawing these:

let R = the number of teeth in the large gear and r = the number of teeth in the small gear

The equation to calculate the number of petals (the protruding curves on the outside of the spiral) is the least common multiple (LCM) of the outer and inner gear numbers (R and r) divided by the inner gear number (r). Written as an equation it is LCM(R, r)/r.

Example 1:

Using an inner gear with 12 teeth and an outer gear with 34 teeth.

This would mean r = 12 and R = 34.

The formula is LCM(R, r)/r so we start by finding the least common multiple of 34 and 12. This can be done by simply googling "LCM(12,34)", but there's also a simple technique to do this by hand.

Start by finding the prime factorization of both numbers. The prime factorization of a number is the single set of prime numbers whose product is the number. If you're not familiar with the concept this Kahn Academy video should help: https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-prime-factorization-prealg/v/prime-factorization

The prime factorization of 12 is 2*2*3 and the prime factorization of 34 is 17*2

To find the least common multiple you multiply the numbers and divide by the duplicated factors. This gives:

(2*2*3*17*2)/2. You can also think of this as dividing by the greatest common denominator. We divide by 2 because 2 is the only factor in the prime factorization of 34 and 12.

(2*2*3*17*2)/2 = 204 and so 204 is the LCM of 12 and 24. Recall the equation is LCM(R, r)/r. So we now divide 204 by 12 (r) and get 17 meaning the design will have 17 petals.

The following is not helpful to drawing spirals. It is just my explanation of the formula for those who are interested from a math perspective.

This formula is derived by considering 2 main ideas:

1. A petal occurs each time the inner wheel rotates
2. All wheel combinations will eventually loop meaning no new petals will be created

Each rotation of the small gear moves r teeth along the larger gear. Once the gear has rotated enough times such that the # of rotations * r > R it will have gone around the big gear more than once. Only once the # of rotations * r is a multiple of R will the gear have returned to the exact place it started. In other words, we're looking for when r * an integer = R * an integer which is what the LCM is. The LCM gives the total number of teeth which need to be traveled so dividing by r gives the # of times the inner gear goes around which is the number of petals.

Additionally dividing by R instead of r gives the number of times the inner wheel goes around the outer wheel which is useful for knowing when you've completed the spiral.

Choose the gears you want to use. It is not required, but it does make the drawing easier to use tape. Fold the tape so that it is sticky on both sides and place it beneath the outer wheel so it's fixed to the paper. Ensure it doesn't interfere with the gear teeth. Now you're ready to draw!

Once you finish the spiral you can shift the gear over one and change colors to have a slightly offset different color spiral. This gives a fantastic effect!

Using colored pencils does an ok job consistently (as long as they're kept sharp), but pens give a more vibrant design.

If you'd like to use this idea for a lesson please reach out to me because I'd love to talk about it! Some of the things I'll teach are:

• What the polar coordinate system is and how to use it
• How to find the LCM of two or more numbers
• How the formula for the number of petals is derived

The questions I plan to use for discussion with the students are:

• How is the cartesian coordinate system extended into 3D?
• How would the polar coordinate system be extended to 3D?
• What are the variables which affect spirographs?
• What would make the corners sharper?
• How can you tell how many petals there will be based on the wheel size?

Thank you for reading!