The Mathematics of Spirograph Curves
The Spirograph toy, patented by British engineer Denys Fisher in 1965, produces curves that mathematicians call roulettes — the path traced by a point fixed to one curve as it rolls along another. The two types produced by Spirograph gears are hypotrochoids (inner rolling gear) and epitrochoids (outer rolling gear). Both families have been studied since the 17th century and appear in optics, planetary gear systems, and the Mandelbrot set.
Parametric Equations
For a small circle of radius r rolling inside a fixed circle of radius R, with the pen at distance d from the rolling circle's centre:
Hypotrochoid: x(t) = (R−r)·cos(t) + d·cos((R−r)/r · t) y(t) = (R−r)·sin(t) − d·sin((R−r)/r · t)Epitrochoid: x(t) = (R+r)·cos(t) − d·cos((R+r)/r · t) y(t) = (R+r)·sin(t) − d·sin((R+r)/r · t)Period: T = 2π · r / gcd(R, r)
The curve closes after r / gcd(R, r) full rotations of the inner gear. The HUD shows this count as "Turns". For simple ratios like R=5, r=3 (gcd=1), the curve closes after exactly 3 turns, drawing a 5-lobed rose. For R=6, r=4 (gcd=2), it closes after only 2 turns, giving a 3-lobed figure.
Preset Parameter Reference
| Preset | R | r | d | Type | Turns to Close | Characteristic |
|---|---|---|---|---|---|---|
| Classic Rose | 5 | 3 | 4.0 | Hypo | 3 | 3-petal rose with rounded ends |
| Five-Point Star | 7 | 2 | 6.0 | Hypo | 2 | 5-pointed star / pentagram shape |
| Daisy Seven | 9 | 2 | 7.0 | Hypo | 2 | 7-petal daisy pattern |
| Cardioid | 4 | 4 | 4.0 | Epi | 1 | Heart-shaped cardioid (r = R special case) |
| Inner Loops | 7 | 4 | 3.0 | Hypo | 4 | Inner loops with 7 arms |
| Galaxy Web | 11 | 3 | 8.0 | Hypo | 3 | 11-arm complex web pattern |
Special Cases & Named Curves
Deltoid (Hypo, d=r, R=3r)
A 3-cusped hypocycloid with cusps forming an equilateral triangle. The envelope of the Simson line for a triangle inscribed in a circle.
Astroid (Hypo, d=r, R=4r)
A 4-cusped hypocycloid: x=R·cos³(t), y=R·sin³(t). Appears as the envelope of a line of constant length sliding along two perpendicular axes.
Cardioid (Epi, d=r, R=r)
The simplest epitrochoid: a heart-shaped curve with one cusp. Forms the main cardioid of the Mandelbrot set and the polar pattern of cardioid microphones.
Nephroid (Epi, d=r, R=2r)
A 2-cusped epicycloid resembling a kidney. Appears as the caustic in the bottom of a tea cup (the bright curved line formed by reflected light).
Rhodonea / Rose Curves
When d < r in a hypotrochoid, the petals don't cross the origin. Increasing d beyond r creates inner loops. At d = |R−r|, the curve passes through the origin.
Wankel Rotary Engine
The rotor in a Wankel engine traces an epitrochoid. The rotor's apex traces the inner housing shape, which is a 2-lobed epitrochoid with specific gear ratios.
Period & Closure: Number Theory Connection
The number of inner-gear rotations before the curve closes equals r / gcd(R, r). This is a consequence of the Euclidean algorithm: the gear returns to its starting orientation only when the total rotation is a multiple of 2π for both the outer and inner angles simultaneously. In terms of the toy's gear teeth, if the outer ring has R teeth and the inner gear has r teeth, the curve closes after r / gcd(R,r) full revolutions of the inner gear — exactly when the teeth re-mesh at their original starting position.
Turns to close = r / gcd(R, r) = lcm(R,r) / RLobes / petals ≈ (R−r) / gcd(R, r) [for hypotrochoid, d near r]
Curriculum Connections
| Level | Topic | Connection |
|---|---|---|
| GCSE Maths | Coordinates, circles, loci | Hypotrochoids as parametric loci of moving points |
| A-Level Maths | Parametric equations, trigonometry | Differentiating x(t), y(t); finding cusps (dx/dt = dy/dt = 0) |
| A-Level FM | Complex numbers, polar curves | Expressing roulettes in polar form; r = a + b·cos(nθ) |
| IB HL Maths | Calculus, vectors | Arc length integral ∫√(ẋ²+ẏ²)dt; curvature κ = (ẋÿ−ẏẍ)/|v|³ |
| University | Differential geometry, group theory | Roulettes as orbits of the rolling group; Klein's Erlangen programme |
| Engineering | Gear design, engine mechanics | Wankel rotor profiles; cycloidal gear tooth profiles |