HYPOTROCHOID

Category: kinetic light installation
Location: New Zealand
Size: ø5m H
Material: machined aluminium, motors, electronics, light source, caron fibre, programming
Year: 2026

HYPOTROCHOID traces arcs of light through three independently rotating armatures, creating geometric figures that are captured over exposures of up to ten minutes. Extending the mathematical logic of the hypotrochoid, the work uses software to identify over 353,000 combinations that produce closed forms. The contrast between a precise digital machine and imperfect analogue film reveals how motion, time, and light are transformed into a single photographic image.

ENTER THE PLAYGROUND

Pattern solver calculates all possible speed and direciton combinations

light scribes arcs through space, leaving traces of thier past positions

HYPOTROCHOID scribes arcs of light visible to the naked eye. Three armatures turn on three independent motors, a light source at the end of the last armature cuts through the air. What the eye follows is an arc. What the camera holds is the whole figure — accumulated in film grain one frame over as long as ten minutes. The apparatus performs. The photograph is the work.

A hypotrochoid is the curve traced by a point fixed to a circle rolling inside a larger one. Whether it closes is a question of arithmetic: rational ratio and the line returns to where it began; irrational and it never does. Three nested rotations extend the curve past its classical two-circle case, and the rule survives intact. Whole ratios close. Everything else leaves the line open ended and incomplete.

HALO XL, the preceding work, held its ratios to prime divisibles and offered 18 closed geometries — a set small enough to enumerate by hand. HYPOTROCHOID lifts that constraint and hands the problem to software. Each armature runs to 5.9rpm, stepped in decimal increments, independently reversible. A pattern solver searches that space for the combinations that close, and returns some 353,215 of them.

The ten-minute cap is not arbitrary. At 0.1rpm — the slowest step the machine will hold — one revolution takes exactly ten minutes. The exposure ceiling is the period of the slowest case, which is to say the shutter is open for precisely as long as the geometry requires and not a second longer. The constraint on the film and the constraint on the mechanism are the same constraint.

The initial captures are digital - as well as recorded on 120 roll film — 6x7cm frames on an Asahi Pentax 6x7 MLU built in 1976, fifty years old this year, and 6x17cm frames on a 3D printed panoramic body carrying a Schneider Super-Angulon 75mm large-format lens. Real chemistry. Ten exposures to a roll on the 6x7, four on the 6x17. The dichotomy is intentional: the machine is digital, precise and incapable of error; the film is analogue, fallible and susceptible to every stray photon that reaches it.

Reciprocity does the rest of the work. Film obeys the reciprocity law — exposure is intensity multiplied by time — only across a middling range; hold the shutter open for ten minutes and the emulsion falls behind, requiring far more light than the arithmetic says it should. The line and the site are affected differently. A moving source writes each point of the trace in a single bright instant; the bush, the containers and the sky accumulate across the full ten minutes and take the full penalty. One frame, two halves of the same response curve.

Sited in bush, on black sand, in a container yard and inside Silo 6. The physics is identical in all four. What changes is what stands behind the line — fern and fallen timber, the Tasman, stacked Matson boxes under sodium light, poured concrete.