If I’m being totally honest, math was never my favorite class. 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.Īdditionally 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.Hate math class? Well, maybe this clever Japanese Twitter user might change your mind! In other words, we're looking for when r * an integer = R * an integer which is what the LCM is. Only once the # of rotations * r is a multiple of R will the gear have returned to the exact place it started. 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.

All wheel combinations will eventually loop meaning no new petals will be createdĮach rotation of the small gear moves r teeth along the larger gear.

A petal occurs each time the inner wheel rotates.

This formula is derived by considering 2 main ideas: It is just my explanation of the formula for those who are interested from a math perspective. The following is not helpful to drawing spirals. So we now divide 204 by 12 (r) and get 17 meaning the design will have 17 petals. We divide by 2 because 2 is the only factor in the prime factorization of 34 and 12. You can also think of this as dividing by the greatest common denominator. To find the least common multiple you multiply the numbers and divide by the duplicated factors. The prime factorization of 12 is 2*2*3 and the prime factorization of 34 is 17*2 If you're not familiar with the concept this Kahn Academy video should help: The prime factorization of a number is the single set of prime numbers whose product is the number.

Start by finding the prime factorization of both numbers. This can be done by simply googling 'LCM(12,34)', but there's also a simple technique to do this by hand. The formula is LCM(R, r)/r so we start by finding the least common multiple of 34 and 12. Using an inner gear with 12 teeth and an outer gear with 34 teeth. Written as an equation it is LCM(R, r)/r. 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). Let R = the number of teeth in the large gear and r = the number of teeth in the small gear Here's some math relevant to drawing these: 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! You can also have 'negative' wheel sizes: 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: What's particularly great about this simulation is the ability to create spirals which aren't physically possible. Hitting play on c begins to draw the spiral. 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. Right click the body in the bodies menu and select save as mesh. The model should then be ready for export. 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 center circle is mostly unimportant since it will just draw a circle. 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. Double click the text and change the text label to the same value as teethNum. In the timeline right click the last sketch then select edit feature. The label text will need to be adjusted manually since parameters can't be used in text. Deselect all faces then select only the center face for the inner wheel or everything outside the the inner circle for the outer wheel. When it does, right click the first extrusion in the timeline and select edit feature. Change the parameter teethNum to the number of teeth you want on your gear.

First, click the modify drop down menu then the change parameters option.

Desmos graph art cake