Math becomes
mesmerizing art.
A hypotrochoid drawing app where GCD and LCM decide the pattern. Tap once and the math draws a closed curve, all the way home.
From a children's toy to top-tier mathematics.
In 2022, Spirograph Maker was acknowledged as a research tool in an algebraic combinatorics paper from the University of Washington — and the paper's title itself elevates Spirographs into a central concept.
The 1965 gear-and-pen toy, faithfully digital.
Spirograph Maker turns the classic gear-and-pen toy into a tap-driven app for iOS and Android. Under the surface the curves are hypotrochoids and epitrochoids — the same parametric family that high-school math calls cycloids. Pick a gear, tap once, and watch the closed orbit bloom into a mandala in seconds. No drawing skill required.
9 years on the App Store · 4.7★ (23 reviews) · 110,000+ downloads · 100+ countries · iOS 18.6+ / Android 8+ · Free with optional ad-removal IAP.
Closed by ratio. Petalled by GCD.
The patterns are not random. The greatest common divisor (GCD) and least common multiple (LCM) — taught in middle school — directly decide how many revolutions a curve takes before it closes, and how many lobes it carries.
x(t) = (R − r) cos t + d cos((R − r) / r · t)y(t) = (R − r) sin t − d sin((R − r) / r · t)
- HypotrochoidInner-gear mode: a circle of radius r rolls inside a fixed circle of radius R, and a point at distance d from its center traces the curve.
- EpitrochoidOuter-gear mode: the same construction, but the inner circle rolls on the outside of the fixed circle.
- Closing conditionIf the gear ratio is the irreducible fraction n/m, the curve closes after m / gcd(n, m) revolutions.
- Number of lobesn / gcd(n, m) — the petals you count in the finished mandala.
A rare topic that bridges middle-school GCD/LCM and high-school cycloid / parametric equations in a single visual.
What the app does, in detail.
Auto-draw mode
One tap and the line traces the entire closed orbit, smoothly, in seconds.
Inner + outer gears
Hypotrochoid (inner gear) and epitrochoid (outer gear) modes — both classical curves.
Custom gear sizes
Adjust R, r, and d freely — every ratio is a new pattern.
10+ colors & 4-point gradient
Pick a palette or set four corner colors for living, shifting gradients along the curve.
Background & pen width
Light, dark, and custom backgrounds. Thin or bold pen strokes.
Undo / Redo
Step back and try a different gear ratio without losing earlier work.
Save to Photos
Export to the Photos library at 2048 × 2048 px for wallpapers and prints.
Share to socials
One-tap share to Instagram, X, and any iOS / Android share extension.
iPad-optimized
Universal binary. iPad's larger canvas shows the fine spiral detail beautifully.
Offline drawing
Drawing and saving work without an internet connection. Network is only used for ads and anonymous analytics.
Three taps from gear to gallery.
Pick gear sizes
Choose inner-gear (hypotrochoid) or outer-gear (epitrochoid) mode and the R, r values that decide how many lobes the curve will have.
Tap to auto-draw
Tap once. The line traces the closed curve and stops the moment it returns home — exactly after m / gcd(n, m) revolutions.
Save or share
Save to Photos, share to socials, or undo and try a different gear ratio.
Where Spirograph Maker fits.
- Bedtime wind-down. A two-minute drawing session quiets the mind before sleep.
- Math class & homework. A visual way to feel GCD, LCM, and parametric curves.
- Independent research projects. Pattern-counting and ratio experiments make a clean self-study project.
- Kids' creative play. Children pick the colors; the math handles the line.
- Original wallpapers. Save and set as lock-screen or home wallpaper.
- SNS art posts. Every output is unique — easy material for daily art posts.
- Focus breaks. A short visual reset between Pomodoros.
The people we draw for.
Also in use in Japanese schools — bulk downloads cluster around the start of the school year and the run-up to summer break.
At a glance.
- Since
- April 2017 · 9 years
- Platforms
- iOS 18.6+ · Android 8+
- Stack
- Swift · Objective-C · Kotlin
- iOS 26
- Liquid Glass ready
- Ads
- AdMob · UMP · ATT compliant
- Price
- Free · optional IAP to remove ads
- Cited by
- arXiv:2207.06508
- Operator
- KYWorks
Free. No sign-up.
Frequently asked.
Why do spirograph patterns close into a flower?
Because the ratio of gear sizes R/r determines a closed orbit. If the irreducible ratio is n/m, the curve closes after m / gcd(n, m) revolutions and forms n / gcd(n, m) lobes. GCD and LCM — taught in middle school — directly govern the shape.
What math is behind the curves?
Hypotrochoid (inner gear) and epitrochoid (outer gear). x(t) = (R−r) cos(t) + d cos((R−r)/r · t), y(t) = (R−r) sin(t) − d sin((R−r)/r · t). The same parametric curves you learn as cycloids in high-school math.
Is it really free?
Yes. The app is free with banner ads. An optional one-time in-app purchase removes the ads — every feature is unlocked on the free tier.
Does it work offline?
Drawing and saving work entirely offline. The only network traffic is ads (Google AdMob, Meta Audience Network mediation) and anonymous analytics / crash reports (Firebase). No artwork ever leaves your device.
Can I use it for math class or homework?
Yes — many users in Japan pick it up around the start of the school year or summer break. It pairs naturally with units on GCD/LCM, cycloids, and parametric equations.
What data is recorded?
Anonymous crash and analytics events via Firebase, plus ad identifiers granted by iOS ATT / Android UMP only when the user consents. No name, email, or location is collected.
How high is the print resolution?
Saved images are 2048 × 2048 px — enough for postcard-size prints and Instagram-quality posts.
What languages are supported?
English and Japanese. The interface and both App Store / Play Store listings are localized.