Spirolateral 003

Twenty-four spirolaterals of increasing order, laid out as a sheet of specimens.

What a spirolateral is

Walk forward one unit, turn. Walk forward two units, turn. Three, turn. Up to n, then start again at one. Keep going.

If the turn is a fixed angle, the walk eventually returns to its start with its original heading, and the figure closes. Which n and which angle you choose decides the shape — and whether it closes at all.

Frank Odds named and catalogued these in 1973. They are a very small idea with a very large output space.

Step 01

The first few figures on the sheet, orders nine to thirteen.

The turn is not always the same way

This variant adds one thing to the classic rule: the turn direction depends on the step number.

if (count == a || count == b || count == c)
    t.Left(angle);
else
    t.Right(angle);

Three step numbers are singled out, and on those the walk turns left instead of right. That breaks the rotational symmetry the plain rule produces and gives each figure a distinct character — which is what makes a sheet of twenty-four of them worth laying out.

Step 02

Two rows.

Step 03

Three rows. The higher the order, the longer the walk runs before the turn pattern repeats, and the more area the figure encloses.

Step 04

Four rows.

The layout is composed, not generated

private static readonly (float X, float Y, int N, float Side)[] Spirals =
[
    (-90f, -90f, 9, 2.0f),
    (-50f, -70f, 11, 1.7f),
    …
];

Twenty-four entries, each giving a position, an order and a step length. Nothing computes them. The positions are irregular — figures are nudged to fit around their neighbours' outlines — and the step length falls as the order rises so that a forty-order figure ends up the same size on the page as a nine-order one.

It is a specimen sheet, arranged by eye, and the arrangement is as much the piece as the figures are.

Step 05

The whole sheet: orders nine through forty.

Five renderings

Rendering	Figures	Ink	Turns
Cyber Cyan	24	cyan on black	90°
Solar Gold	—	amber on near-black	90°
Matrix Green	—	neon green on near-black	90°
Prism Violet	—	violet on near-black	60°
Museum	—	cyan on black, warm mat	90°

Cyber Cyan, Solar Gold, and Matrix Green share the same 90° geometry — only palette differs. Prism Violet uses 60° turns; the asymmetric turn pattern produces expanding walks rather than compact loops, giving the sheet a different character. Museum presents Cyber Cyan with a warm-cream mat border — the same figure, gallery-framed.

Determinism

Nothing is seeded and nothing is random. Every figure is a fixed turtle walk from a fixed starting position.

Origin

Ported from a Turtletoy sketch by rupertxrussell.

Progression 05

Variant 05