GART

Parallel Curves

A bundle of strands that runs straight, turns a tight corner, and runs on, every strand the same distance from its neighbours the whole way round.

Parallel Curves: A bundle of strands that runs straight, turns a tight corner, and runs on, every strand the same distance from its neighbours the whole way round.Parallel Curves: A bundle of strands that runs straight, turns a tight corner, and runs on, every strand the same distance from its neighbours the whole way round.

About this pattern

Parallel curves · Cyril Stanley Smith's quarter-circle Truchet, 1987

Parallel Curves, truchet tiles, drawn by GART: A bundle of strands that runs straight, turns a tight corner, and runs on, every strand the same distance from its neighbours the whole way round.
Parallel Curves · CC BY 4.0

Every strand here is the same path, offset sideways by a fixed distance, what geometers call a parallel curve. On a straight run that is trivial. On a turn the inside strand has to take a smaller circle than the outside one, around the same centre, which is why the bends read as concentric and why the innermost strand can tighten to almost nothing.

Each turn is one of Smith's quarter-circle Truchet tiles drawn several strands deep; the walk between them avoids its own track, except that it may cross a straight run at right angles and pass over it.

TopicsParallel curve (opens in a new tab)Truchet tiles (opens in a new tab)

Behind the mathematics

Cyril Stanley Smith1903–1992

Truchet's tile as quarter circles, 1987

Metallurgist and historian of science who rediscovered Truchet in 1987 and replaced his diagonal with two quarter circles, turning the tiles into continuous paths.