A Spirograph is a geometric drawing instrument that traces roulette curves — specifically hypotrochoids and epitrochoids — by rolling a small toothed wheel inside or outside a fixed toothed ring while a pen sits in an offset hole on the wheel. The curve's shape depends entirely on the tooth-count ratio between ring and wheel, plus the pen's radial offset from the wheel centre. Draughtsmen, mathematics teachers, and pattern designers use it to generate repeatable, mathematically exact ornamental curves without freehand skill. Denys Fisher's 1965 toy version sold over 30 million units worldwide.
Spirograph Hypotrochoid Interactive Calculator
Vary the fixed ring size, rolling pinion size, and pen offset to see the hypotrochoid curve, closure laps, lobe count, and offset sharpness update live.
Equation Used
The calculator uses the worked hypotrochoid equations for a pinion rolling inside a fixed ring. R is the fixed ring size, r is the rolling pinion size, and d is the pen-hole offset from the pinion center. The closure outputs reduce the rolling spin ratio (R-r)/r using gcd(R-r,r).
FIRGELLI Automations - Interactive Mechanism Calculators.
- Inside-the-ring rolling creates a hypotrochoid.
- Pure rolling contact with no slip.
- R, r, and d use the same pitch or length units.
- R and r are treated as integer pitch/radius counts for closure.
How the Spirograph Actually Works
A Spirograph works on pure rolling contact. You pin a fixed ring gear to the paper, drop a smaller pinion gear inside (or outside) the ring, push a ballpoint through one of the offset pen holes on the pinion, and walk the pinion around the ring. As the pinion rolls without slipping, the pen traces a roulette curve — the locus of a point on a circle rolling along another circle. Inside-the-ring tracing gives you a hypotrochoid. Outside gives you an epitrochoid. Same instrument, two different curve families, decided only by which side of the ring you put the pinion on.
The shape of the curve comes from three numbers: the ring tooth count R, the pinion tooth count r, and the pen-hole offset d from the pinion centre. The ratio R/r decides how many petals or lobes the curve has and whether the curve closes after one trip or twenty. If R/r reduces to a low fraction like 96/64 = 3/2, the curve closes in two laps with 3 lobes. If R/r is something awkward like 96/63, you get a long, dense, near-non-repeating pattern that takes 63 laps to close. The pen-hole offset d controls how sharp or rounded the lobes are — small d gives a near-circle, d close to r gives sharp cusps.
If the rolling contact slips even once, the whole pattern shifts and you'll see a doubled or smeared lobe where the slip happened. That's the single most common failure mode. The teeth must engage cleanly, the paper must not move, and you must keep the pen pressure consistent. The original Kenner/Fisher gears use a 0.8 mm tooth pitch and a small clearance — push too hard sideways on the pen and the pinion lifts a tooth, instantly killing the geometry. A worn pin or a soft cardboard backing under the paper is enough to introduce slip. Tolerances on tooth-pitch matching matter — a pinion from a different manufacturer with a 0.85 mm pitch will not mesh cleanly with a 0.8 mm ring and the curve will drift.
Key Components
- Fixed Ring (annular gear): The stationary outer guide. Has internal teeth (for hypotrochoid work) and often external teeth (for epitrochoid work). Original Denys Fisher rings carry tooth counts of 96, 105, and 144 to give a useful spread of ratios with the supplied pinions.
- Rolling Pinion (wheel): The small toothed disc that rolls against the ring. Each pinion has multiple pen holes drilled at different radii — typically 6 to 8 holes per wheel. Standard Denys Fisher pinion tooth counts include 24, 30, 32, 36, 40, 45, 50, 52, 56, 60, 63, 64, 72, 75, 80, and 84.
- Pen hole: A precisely-positioned bore on the pinion that accepts a fine ballpoint. The radial distance d from pinion centre to pen-hole centre is the third parameter that defines the curve. Hole position tolerance is roughly ±0.1 mm — beyond that, lobe symmetry visibly degrades.
- Anchor pins or adhesive putty: Holds the ring stationary against the paper. The original 1965 set used straight pins through cardboard; modern Hasbro Super Spirograph kits use removable putty pads. Any movement of the ring during a draw produces a stepped artefact in the curve.
- Backing board: A firm cardboard or foam-core sheet under the paper. Without it the pins won't hold and the paper distorts under pen pressure, which translates directly into pinion slip and lost teeth.
Industries That Rely on the Spirograph
Spirographs survive in three working niches: education, decorative pattern generation, and security printing. The geometry is too constrained to compete with CAD for general drafting, but where you need a repeatable, parameterised closed curve drawn by hand, nothing else is faster. The same hypotrochoid math powers anti-counterfeit guilloché patterns on banknotes and share certificates, drawn historically on geometric lathes that are mechanically identical to a Spirograph scaled up and motorised.
- Education: UK Key Stage 3 mathematics classes use Hasbro Spirograph Deluxe Design Sets to teach ratio, lowest common multiple, and parametric curves — the lap-count needed to close a curve is a direct LCM exercise.
- Security printing: Guilloché patterns on Bank of England notes and the US Treasury share certificates are large-scale hypotrochoids and epitrochoids — the original geometric lathes at De La Rue and the American Bank Note Company use the same rolling-circle principle as a Spirograph.
- Toy and giftware: Denys Fisher's original 1965 Spirograph and the current PlayMonster Spirograph Deluxe Design Set are the dominant retail products — over 30 million units sold to date.
- Watchmaking and jewellery engraving: Decorative guilloché on watch dials from manufacturers like Breguet and Vacheron Constantin is cut on rose engines that produce hypotrochoid patterns identical in geometry to Spirograph output, just at sub-millimetre line spacing.
- Generative art and design: Pattern designers at studios like House Industries use physical Spirograph kits as a fast prototyping tool before moving designs into Adobe Illustrator using a parametric hypotrochoid script.
- Mathematics outreach: The Mathematikum museum in Giessen has a 1.2 m diameter floor-mounted Spirograph used by visitors to draw curves with a 60 cm pinion — a direct teaching demonstration of roulette geometry.
The Formula Behind the Spirograph
The hypotrochoid parametric equations tell you exactly where the pen is at any rolling angle θ. They matter because before you commit ink to paper, you want to know two things: how many lobes will this combination produce, and how many laps does it take to close. At the low end of useful ratios — say R/r = 2 — you get a degenerate near-line and the pattern closes in one lap. At the nominal range most users actually work in, R/r between 1.3 and 3, you get 3 to 20 lobes and the curve closes in 1 to 10 laps, which is the sweet spot for paper size and patience. Push to R/r ratios that don't reduce, like 96/63, and the curve takes 63 laps to close — beautiful but slow, and any tooth slip across that many revolutions ruins it.
y(θ) = (R − r) × sin(θ) − d × sin((R − r) / r × θ)
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| R | Radius of the fixed ring (proportional to ring tooth count) | mm | in |
| r | Radius of the rolling pinion (proportional to pinion tooth count) | mm | in |
| d | Pen-hole offset from the pinion centre | mm | in |
| θ | Rolling angle of the pinion centre about the ring centre | rad | rad |
| Nlaps | Laps required for the curve to close = r / gcd(R, r) | — | — |
Worked Example: Spirograph in a luthier engraving a rosette on an acoustic guitar soundboard
A luthier in Granada is laying out a decorative rosette around the soundhole of a classical guitar. He wants a closed hypotrochoid with clean lobes that fits inside an 84 mm outer diameter ring around the 86 mm soundhole. He has a Hasbro Super Spirograph set and picks the 96-tooth ring and a 36-tooth pinion. The pen-hole offset d is 12 mm. He needs to know how many lobes he'll get, how many laps to close the curve, and what the maximum radial extent of the pattern will be — he cannot overshoot the binding channel.
Given
- R = 48 mm (ring radius, scaled from 96 teeth)
- r = 18 mm (pinion radius, scaled from 36 teeth)
- d = 12 mm
- Tooth ratio = 96 / 36 —
Solution
Step 1 — reduce the tooth ratio to find lobe count and lap count. gcd(96, 36) = 12, so the reduced ratio is 96/36 = 8/3.
Step 2 — at the nominal pen offset of d = 12 mm, compute the maximum radial extent of the curve from the ring centre:
Step 3 — at the low end of the practical offset range, d = 4 mm (innermost pen hole on a typical 36T pinion), the lobes are barely-rounded bumps on a 30 mm radius circle. rmax = 34 mm, rmin = 26 mm — pattern is shallow, not visually distinct enough for a guitar rosette.
Step 4 — at the high end, d approaches r itself (d = 17 mm, outermost hole), and the lobes sharpen into near-cusps. rmax = 47 mm, rmin = 13 mm. The cusps look elegant but the radial swing now nearly fills the 48 mm ring radius, leaving 1 mm of margin — too tight for safe engraving on the guitar where the binding channel sits at 43 mm.
Result
The nominal 96/36 setup with d = 12 mm produces a 5-lobed hypotrochoid that closes in 3 laps with a maximum radial extent of 42 mm — a confident 1 mm clearance inside the 43 mm binding channel and a clean visual rhythm. At the low end (d = 4 mm) the lobes are too shallow to read as ornament; at the high end (d = 17 mm) the cusps are gorgeous but the 47 mm reach overshoots the binding channel and the rosette gets cut into during routing. The sweet spot sits squarely at d = 10 to 13 mm. If your finished rosette shows 6 lobes instead of 5, or one lobe is doubled, the cause is almost always tooth slip on lap 2 or 3 — check that the ring has not crept on its putty pads and that the pinion did not lift off when the pen crossed a paper seam. A second common cause is a pinion-pen-hole bore that has worn oversize from years of use, letting the pen wander radially by 0.3 to 0.5 mm and breaking the symmetry between adjacent lobes.
Choosing the Spirograph: Pros and Cons
Spirograph isn't the only way to draw a parametric curve. Compare it to a pantograph (mechanical curve copier) and modern CAD plotting on the dimensions a working designer actually cares about.
| Property | Spirograph | Pantograph | CAD plotter / vector software |
|---|---|---|---|
| Curve families produced | Hypotrochoids and epitrochoids only | Any 2D curve (copies an input) | Any parametric or freeform curve |
| Setup time per pattern | 30 seconds — pin ring, drop pinion | 5–10 minutes alignment | Minutes to hours of CAD entry |
| Repeatability of identical pattern | Excellent — defined by tooth count and offset, ±0.3 mm | Poor — operator-dependent | Perfect — digital file |
| Cost (entry-level) | £15–£30 (Hasbro Deluxe set) | £40–£200 (drafting pantograph) | £0–£600 (Inkscape free, Illustrator paid) plus plotter |
| Maximum drawing diameter | ~140 mm with stock kit, scalable with custom rings | Up to 1 m with large pantograph | Limited only by plotter bed (A0 and beyond) |
| Slip / error mode | Tooth slip ruins curve | Pivot wear introduces drift | Digital — no mechanical error |
| Skill required | Low — children use it | Moderate — alignment skill | High — CAD literacy |
Frequently Asked Questions About Spirograph
The pinion is climbing slightly out of plane on each lap. This happens when pen pressure pushes the pinion downward and one edge of the wheel lifts the opposite edge off the ring teeth — even a 0.2 mm lift is enough to skip half a tooth per revolution. Over 3 laps that compounds into visible sideways drift.
Fix it by reducing pen pressure, switching from a ballpoint to a gel pen which writes with much less downforce, or putting a finger lightly on the far side of the pinion to keep it flat. Worn pen holes that have ovalised also cause this — if the pen rocks in the hole more than 5° off vertical, retire that pinion.
Compute gcd(Rteeth, rteeth) and then Nlaps = rteeth / gcd. Anything above 20 laps becomes tedious and risks accumulated slip — most practitioners stay under 10 laps. The 96-tooth ring with the 32, 36, 48, 60, or 72 pinions all give low lap counts because they share large common factors with 96. The 63-tooth pinion against a 96-tooth ring gives gcd = 3 and 21 laps — drawable, but you'll spend half an hour on it and one mis-step kills it.
Choose epitrochoid when you want lobes that point outward like a flower or gear-tooth profile, and hypotrochoid when you want a star or rosette pattern with inward-pointing cusps. Epitrochoids are also more forgiving on paper position because the pinion can't get trapped — you can pause and re-grip mid-lap, which matters on long patterns.
The downside of epitrochoid work is that the pattern grows larger than the ring itself, so you need more paper area. A 96-tooth ring with a 24-tooth pinion epitrochoid will reach roughly 130 mm in diameter — the hypotrochoid version of the same combination stays inside the 96 mm ring.
Almost always a counting error in the assumed tooth count, not a mechanical fault. Lobes = (Rteeth − rteeth) / gcd for hypotrochoids. If you grabbed a 30T pinion thinking it was 36T, the math changes: 96/30 reduces via gcd = 6 to 16/5 — that gives 11 lobes in 5 laps, and a partial draw stopped after 3 laps will read as a 6 or 7-lobe shape mid-pattern.
Count teeth physically before trusting the moulded label. Several Hasbro pinions have the number printed on one side only and it's easy to grab a 32T thinking it's a 36T. Set them on a known reference pattern first.
Yes, and several artists do. Laser-cut MDF or acrylic ring-and-pinion sets at module 1 or module 1.5 work well up to roughly 800 mm ring diameter. Above that, gear flex becomes a problem — a 1 m acrylic ring will deflect 2 to 3 mm under the pen-arm load and the lobes lose symmetry on the far side of the pattern.
For scales over 1 m, switch to a chain-and-sprocket drive or a friction-roller geometric lathe — that's the same engineering choice industrial guilloché engravers made in the 19th century. Pure gear-rolling stops being practical above about 1.2 m.
Three likely causes that don't involve slip. First, the ring isn't truly fixed — putty creeps under sustained sideways force, especially in warm rooms above 22 °C, and you'll see lobes 4 and 5 of a 5-lobe pattern shifted from lobes 1 and 2. Second, the paper itself shifts on the backing board if the pins aren't through both layers. Third, the pen-hole isn't where you think it is — moulded plastic pinions have hole-position tolerances of ±0.1 to ±0.15 mm, which translates to visibly asymmetric lobes when d is small (under 6 mm).
Diagnostic check: draw the same pattern twice on the same sheet, rotating the ring 180° between draws. If the two patterns overlay perfectly, the pinion is fine and your problem is ring or paper movement. If they don't, the pen-hole is off-centre and that pinion is the culprit.
References & Further Reading
- Wikipedia contributors. Spirograph. Wikipedia
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