3D Printed Spirographs
Introduction: 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.
Supplies
* 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
Step 1: Assembling Desired Files
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.
Step 2: Modifying Fusion360
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. 737
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.
Step 3: Slicing
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.
Step 4: Math!
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. 110