Golden ratio spiral generator
A golden spiral is already drawn: the curve that gets 1.618 times wider with every quarter turn. Press Fibonacci for the famous look-alike made of quarter circles in squares, and download either one as an SVG or a PNG.
Free, no account. The file is read by code running in your browser. It is not uploaded, and nothing about it is stored.
A golden spiral: 3 turns, 1.618 times wider with every quarter turn.
Fine-tune: exact size, turns, line width, colour
- Size
- 100 × 63.89 mm; the page adds half the line’s width all round: 101.5 × 65.39 mm
- Curve
- r = 0.2218 mm × e^(0.3063 × θ), with θ in radians
- Line
- 243 mm long and 1.5 mm wide, in one path
- File
- golden-spiral-3-turns.svg, 4.1 KB
What the golden ratio spiral is
The golden ratio is the number (1 + √5) ÷ 2, about 1.618, written φ. A golden spiral is the logarithmic spiral that grows by that ratio every quarter turn: go a quarter of the way round and you are 1.618 times as far from the middle. Half a turn multiplies the distance by φ², about 2.618, and a full turn by φ⁴, about 6.854.
As an equation, with r the distance from the middle and θ the angle in radians, it is r = a × e^(b × θ) where b is the natural logarithm of φ divided by a quarter turn (π ÷ 2), which comes to 0.3063. The number a only sets the size. Like every logarithmic spiral it crosses each straight line from its middle at the same angle, here about 73°.
The curve has no beginning. Followed inwards it winds round the middle for ever, shrinking by 6.854 times with each turn, so a drawing has to stop somewhere. This page stops after the turns you choose: three turns in, the line is 322 times closer to the middle than where it started, which on a spiral 100 mm wide is under a quarter of a millimetre.
The Fibonacci spiral only approximates it
The picture most people know has squares in it. Draw two squares of side 1 next to each other, a square of side 2 against the pair, then squares of side 3, 5, 8, 13 and 21, each against the long side of everything before it, turning as you go. The sides are the Fibonacci numbers, in which each is the sum of the two before. A quarter circle in each square, joined end to end, makes the Fibonacci spiral.
It is not the golden spiral. From one quarter circle to the next its radius grows by the ratio of two neighbouring Fibonacci numbers: 1, then 2, then 1.5, 1.667, 1.6, 1.625, 1.615. Those ratios close in on 1.618 from both sides and never reach it. And each piece is an arc of a circle, which bends by the same amount all along and then changes abruptly at the joint, where the golden spiral’s bend loosens smoothly the whole way out.
Unlike the golden spiral, the Fibonacci spiral has a definite beginning: the first square. Quarter circles drawn in the squares of a true golden rectangle make a third curve, close to both, which this page does not draw.
How to draw one here
- The page arrives on the golden spiral with three turns. Choose 2 or 4 turns, or type another number under Fine-tune.
- Press Fibonacci to switch to the quarter circles. Choose 6, 8, 10 or 12 squares, and whether the squares are drawn or left out.
- Pick the line (thin, medium or bold) and which way the spiral winds outwards.
- Set the size under Fine-tune: it is the longer side of the spiral’s own box, in millimetres.
- Download the SVG, or the PNG for laying over a photograph or a layout.
What you download
The golden spiral is one stroked path of smooth curves, cut into 32 pieces a turn, with the quarter-turn points lying exactly on the curve: measure from the middle to four neighbouring ones and each distance is 1.618 times the last. Three turns set to 100 mm wide make a file of about 4 KB, 100 × 63.89 mm.
The Fibonacci spiral is written as circular arcs, one A command to each quarter circle, with the squares as a second, thinner path. Eight squares fill a rectangle of 34 by 21 units, so at 100 mm wide it is 61.76 mm tall and its smallest square is 2.941 mm. Both kinds are sized in millimetres, and the PNG has a see-through background.
Using it, and what to doubt
Designers lay the spiral over a photograph or a page as a guide for where the eye should travel, with the tight end on the subject. The PNG is made for that: place it as a layer in any picture editor and turn or flip it to suit. This page does not load your photograph.
Be wary of claims that the curve is everywhere in nature. Shells, storms and galaxy arms are roughly logarithmic spirals, but a logarithmic spiral is golden only when it grows by 1.618 every quarter turn, and nothing obliges a shell to grow at that rate. A spiral that widens by 1.3 a quarter turn, or by 2, is just as logarithmic. Before calling a spiral golden, measure it: the distance from the middle should be 6.854 times greater after one full turn.
Questions
Is the Fibonacci spiral the same as the golden spiral?
No. The golden spiral is one smooth curve that grows by exactly the golden ratio every quarter turn. The Fibonacci spiral is a chain of quarter circles whose radii are Fibonacci numbers, so it grows by a slightly different ratio at each step and only approximates the golden spiral. This page draws both, so you can compare them.
What is the equation of the golden spiral?
In polar form, r = a × φ^(2θ ÷ π), with θ in radians and φ = 1.618. That is the same as r = a × e^(0.3063 × θ). The page shows the equation with the value of a for the spiral it has drawn, in millimetres.
Does it matter which way the spiral winds?
Not to the mathematics: the clockwise and anticlockwise spirals are mirror images of one curve. For a layout it matters a great deal, so the page offers both, and the saved file can be turned or flipped in any editor.
Why is the middle of the golden spiral so small?
Because each turn inwards is 6.854 times smaller than the last. After two turns the curve is 47 times smaller and after three, 322 times. Fewer turns leave an open middle; more turns add coils too small to see.