Spirograph Glossary

The mathematical name for the family of curves produced by a Spirograph. A hypotrochoid is traced by a point on a small circle rolling inside a larger fixed circle. The prefix "hypo-" means below or inside (as in hypothermia). Every standard Spirograph pattern is a hypotrochoid.

The sibling curve to the hypotrochoid. An epitrochoid is traced by a point on a small circle rolling outside a fixed circle. The Spirograph logo is an epitrochoid. "Epi-" means outside or upon (as in epidemic). This visualizer currently draws hypotrochoids only.

The large fixed outer gear. It stays stationary on the paper while the wheel rolls around its inner edge. Rings are identified by their tooth count: the Hasbro Deluxe set includes rings with 96, 105, 144, and 150 teeth. More teeth = larger physical ring = larger drawn pattern.

The small rolling gear that fits inside the ring. The wheel's outer teeth mesh with the ring's inner teeth, so it rolls without slipping. Pen holes at various distances from the wheel's center allow different curve shapes. The Deluxe set includes 18 wheels: 24, 30, 32, 36, 40, 42, 45, 48, 50, 52, 56, 60, 63, 64, 72, 75, 80, and 84 teeth.

The number of lobes, points, or "petals" visible in the finished spirograph pattern. Calculated as lcm(ring, wheel) / wheel. Simple combinations (large GCD) give few petals; coprime combinations (GCD = 1) give the most.

The number of times the wheel must travel around the interior of the ring before the pen returns to its exact starting point and the pattern closes. Equal to lcm(ring, wheel) / ring. A higher number means a more complex, time-consuming drawing.

The ratio of ring teeth to wheel teeth, simplified by dividing both by their GCD. Two combinations with the same simplified ratio produce the same pattern shape at different scales. Ring 96 / wheel 32 and ring 150 / wheel 50 both simplify to 3/1 — a 3-petal pattern.

The smallest positive integer divisible by both the ring and wheel tooth counts. The LCM is the key number: it equals the total number of teeth the wheel travels before the pattern closes, and determines both petal count and rotations. Computed as: lcm(a, b) = (a × b) / gcd(a, b).

The largest number that divides both the ring and wheel tooth counts without remainder. A large GCD means the two gears are highly "related" — the pattern will close quickly and have few petals. A GCD of 1 (coprime) means no early closure; the pattern reaches maximum complexity.

The distance of the pen from the wheel's center, expressed as a fraction of the wheel's radius (0 to 1, or 0% to 100%). Pen ratio 0 draws a circle. Pen ratio 1 (edge) draws the sharpest, most star-like version of the pattern. Values between 0.6 and 0.85 give the classic spirograph aesthetic.

A curve defined by expressing x and y coordinates separately as functions of a single parameter (usually angle θ). Spirograph curves are parametric: x(θ) and y(θ) are both defined in terms of the angle the wheel has turned, rather than as a direct y = f(x) relationship.

A special case of the hypotrochoid where the pen is placed exactly at the wheel's edge (pen ratio = 1.0, or d = r). Specific ring/wheel ratios produce named hypocycloids: R/r = 2 gives a straight line; R/r = 3 gives a deltoid; R/r = 4 gives an astroid (four-pointed star).

In the context of this visualizer, a layer is one complete spirograph curve — a single ring+wheel+pen combination drawn in one color. Multiple layers can be composited on the same canvas to create multi-color designs. Physically, this corresponds to drawing successive patterns on the same sheet of paper without moving the ring.