A spirograph drawing machine decides its petal count with gears, before the pen ever touches the paper. Work out how many times the pen swings in and out for each turn of the paper, write that as a fraction in its lowest terms, and read it off: the top number is the petals, the bottom number is how many trips round the paper it takes to finish. On the Spirograph Drawing Machine that fraction is 7/3, so it draws a 7-petal flower that closes after 3 turns of the paper.
That one rule makes this build a good bridge between a maths lesson and an art one. You can predict the drawing, then crank and check.
How the machine works
The Spirograph Drawing Machine is a level 3 build that takes about 45 minutes, plus a felt pen, paper and some sticky tape. One red hand crank does everything:
- A 12-tooth corner gear on the crank shaft bites into the 28 teeth of a turntable. The paper is taped to a frame on top, so it turns 12/28 of a turn for every crank turn: a bit less than half.
- A second pair of 12-tooth corner gears turns a standing shaft 1 : 1, taking the power round the corner the way bevel gears do.
- On top of that shaft a small crank pulls a 3-hole link, which rocks an 11-hole pen arm on its hinge. Crank, link, arm and frame make a four-bar linkage: the crank goes all the way round, but the arm only swings, about 29° from side to side.
- A pen holder on the end of the arm lets the pen slide up and down, so it stays on the paper while the arm swings it in and out across the middle.
The pen swings once per crank turn. The paper turns slowly underneath. Neither motion alone makes a flower: the swinging pen on still paper draws a short line, and a still pen on turning paper draws a circle. Together they draw loops.
The petal rule
Count everything per turn of the paper.
- The paper turns 12/28 of a turn per crank turn.
- The pen swings 1 time per crank turn.
- So per paper turn, the pen swings 1 ÷ (12/28) = 28/12 times.
- In lowest terms, 28/12 = 7/3.
7/3 means 7 swings for every 3 paper turns. After 3 turns the paper and the pen are both back where they started, at the same moment, so the line joins itself and the flower is done. Those 7 swings are spread evenly round the paper, which is why there are 7 petals and not 2⅓.
It is the same rule people use on the toy spirograph's rings and wheels: divide the ring's teeth by the biggest number that divides both tooth counts, and you get the number of points. Here 28 and 12 share a 4, and 28 ÷ 4 = 7.
If you've read when two gear markers return, the closing part will look familiar: the drawing finishes when the paper and the pen line up again. The petal count is the new part, and it is what the lab tests.
Predict a flower
Interactive: Spirograph petals lab. The Spirograph Drawing Machine’s flower drawn from its real geometry: change the corner gear on the turntable, the pen shaft gears or the link’s hole, predict the petals, then watch the pen draw. Five challenges on why the gears set the petal count and the holes only change the shape.
The lab solves the machine's real linkage and draws the pen's path on the turning paper. Most of the settings are what-ifs, because the kit has corner gears with 12 and 20 teeth but not 14 or 28. Three results surprise most people:
- A 20-tooth gear on the turntable still gives 7 petals. 28/20 = 7/5, so the flower now takes 5 paper turns and the petals overlap more, but the count doesn't change. 12 and 20 both share a 4 with 28.
- Speeding up the pen shaft changes everything. If a 20-tooth corner gear drives a 12-tooth one, the pen swings 5/3 times per crank turn, and the flower jumps to 35 petals over 9 paper turns.
- The link's hole changes the shape, not the count. The booklet's playtime tip says to move the link to a different hole of the arm. In the lab, 4 or 5 holes from the hinge work; at 3 or 6 the link can't close the loop and the crank jams.
Try it as a lesson
The machine reads well as a 3-part activity:
- Predict. Before anyone cranks, work through the rule as a class and write the answer on the board: how many petals, and how many paper turns?
- Draw. Crank slowly and count the paper turns. The build calls for a felt pen, held so its tip just touches the paper. Mark the pen's start point so you can see the line join up.
- Explain the differences. Real drawings won't be perfectly crisp: the pen drags, and gears have a little play. The petal count should still match. If it doesn't, check that the corner gear hasn't slipped a tooth on the turntable.
Then let people change one thing, predict again and draw again. A wall of flowers with their fractions written underneath makes the maths visible.
Before and after
The machine's build page suggests building the Spinning Radar and the Bevel-Gear Windmill first: the radar teaches the turntable and slow gearing, the windmill the corner gears. For more on why a crank-driven arm only swings, the walking legs explainer covers four-bar linkages. And for another way into mechanisms from outside engineering, the cam drum machine turns a shaft into a bar of music.
Interactive: Check yourself. A few quick questions on the ideas in this guide, each with an explanation.
- The paper turns 12/28 of a turn for every crank turn, and the pen swings once per crank turn. How many pen swings happen per paper turn?
- Pen swings per paper turn come out as 7/3 in lowest terms. What does the drawing look like when it is finished?
- You move the link to a different hole of the pen arm, and the crank still turns all the way round. What changes?
- Which gear swap gives a flower with fewer, simpler petals?
