Free Polar Grid Generator

The default is the Archimedean spiral, described by Archimedes in On Spirals around 225 BC. Its rule is that the radius grows in direct proportion to the angle: turn twice as far, get twice as far out. The visible consequence is that the gap between successive turns is constant.

That constant gap is the signature of anything that has been coiled. A rope on a deck. The groove on a vinyl record. A rolled rug, a coil pot, a clock spring, a roll of tape. In every case a strip of constant thickness has been wound, so each turn is displaced outward by exactly that thickness.

The other spiral, switched on by the Fibonacci-growth checkbox, is logarithmic. Its rule is that the radius is multiplied by a fixed factor each turn rather than increased by a fixed amount. Here that factor is φ², or 2.61803, which is the golden ratio squared.

Why φ² and not φ? Because the golden spiral is conventionally defined as growing by a factor of φ per quarter turn, which compounds to φ⁴ per full turn — that is the very tight construction drawn inside a golden rectangle. The φ²-per-turn version is the looser, more open curve that is usually what people mean when they describe a shell as a golden spiral, and it is the one that lies on real specimens far more often.

At the default of four turns, the growth is dramatic: the outermost turn is about 17.9 times the width of the innermost. That is not a subtle bias in the curve; it is the whole character of it.

A caution worth stating plainly, because the internet is not careful about it. Real nautilus shells are logarithmic spirals, but their growth ratio is typically around 1.31 to 1.33 per turn, not 2.618. The claim that a nautilus is a golden spiral has been measured repeatedly and is not supported. It is a logarithmic spiral, which is the interesting and true fact; the golden part was added later by people who wanted the story tidier.

The same caution applies to sunflowers, pine cones and pineapples. Those genuinely do involve the golden ratio, but through a different mechanism — the angle between successive seeds is the golden angle of about 137.5°, which produces the interlocking spiral families you can count. The counts are Fibonacci numbers. That is a real and well-documented phenomenon, and it is not the same as the outline being a golden spiral.

Which brings up the honest use of this overlay: comparison. Lay the spiral on a photograph of an actual shell and adjust the turns and the growth mode until it fits, or discover that it does not. That is a more useful five minutes than reading any number of articles asserting one thing or another.

For composition, the spiral is a path rather than a placement. A rule-of-thirds grid gives you four points to put things at. A spiral gives you a route for the eye to follow, and its use is in leading a viewer from the entry point of a picture to its subject.

That makes it well suited to subjects that already have rotational movement — a staircase, a coiled rope, a wave, a crowd curving round an obstacle — and poorly suited to still, frontal subjects, where it imposes a motion the picture does not have.

The turn count changes the character more than any other setting. One turn is a broad arc, barely a spiral, useful as a gentle lead-in. At ten turns the figure is a tight coil that reads as a target or a vortex. Four, the default, is where the curve is legible as a spiral while still crossing enough of the frame to be useful.

One technical limit to know: the curve is sampled at a fixed 260 points regardless of turn count. At four turns that is 65 points per turn and the curve is smooth. At ten turns it is 26 per turn, and the innermost coils show visible faceting. If you are exporting a tight spiral for reproduction, check the inner turns before you commit.

The rings and spokes serve the spiral rather than competing with it. They are the ruled paper: their job is to let you say where on the curve you are, so that when you transfer the spiral to another surface you can register it. For that purpose more is better, up to the point where the grid starts to obscure the subject.

Polar grids have uses beyond spirals. Any data with an angular dimension wants one — wind roses, radar plots, circular histograms, the astronomical charts that plot position by right ascension and declination. Pottery decoration, wheel-thrown work and turned wood all need the circumference divided honestly.

For an ornamental figure rather than a measurement, the mandala grid generator divides the circle into equal segments, and the golden spiral generator draws the growth spiral on its own. For crafts, the polar grid is the layout tool for anything circular: a round rug, a doily, a mandala quilt, a spiral-bound basket. In all of those the construction proceeds outward from a centre, and having rings and divisions marked in advance is the difference between a piece that closes and one that gradually goes out of true.

The overlay draws over your own image rather than exporting a fixed template because the centre placement is the first decision. A spiral centred in the frame reads as an emblem; a spiral whose centre sits off in one corner reads as a movement through the picture.

Export sizes itself from the paper format and the DPI you choose, not from your screen. SVG keeps the spiral as a single path you can re-stroke, recolour or use as a motion guide in an animation tool. The polar and spiral overlay page separates the two, beside the rest of the specialty grid category.

Everything runs in your browser. Images you load are never uploaded, because there is no server for them to reach. The drawing, the transform and the export all happen on your own machine.