Mystical systems
Logarithmic Spiral
Direct answer
Logarithmic Spiral: A logarithmic spiral is a curve that widens by the same ratio with every equal turn, so that it keeps its shape at every size. The golden spiral is one member of the family.
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At a glance
- Also called
- golden spiral, equiangular spiral, nautilus spiral, spira mirabilis
- Found in
- Mystical Systems
- Related
- Golden Ratio, Fibonacci Sequence, Spiral, Spiral Symbol
A logarithmic spiral is a curve that widens by the same ratio with every equal turn, so that it keeps its shape at every size. The golden spiral is one member of the family.
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What is a logarithmic spiral?
In a logarithmic spiral the distance from the centre is multiplied by a fixed number for each equal angle turned. Any part of it, enlarged, lies exactly over any other part, and a line from the centre always crosses the curve at the same angle, which gives it its other name, the equiangular spiral. There are endlessly many such spirals, one for each rate of growth.
The golden spiral is the one that grows by the golden ratio, about 1.618, every quarter turn. The familiar drawing of quarter circles inside squares sized by Fibonacci numbers is a close approximation to it. A nautilus shell is a logarithmic spiral too, because the animal grows without changing shape, but measured shells widen by roughly a third each quarter turn and not by 1.618.
On the tree of the hidden arts: Sacred geometry.
What sets Logarithmic Spiral apart
The curve has one defining habit: grow it, shrink it or turn it, and the same curve comes back. That is why it marks anything that adds to itself at one end in proportion to its size, a shell, a horn, a tusk. It never reaches its centre, winding inward without end. The spiral Archimedes studied is a different thing, with coils evenly spaced like a rolled carpet, and that is the one carved by mistake on Bernoulli's tomb.
- Also called: the equiangular spiral; spira mirabilis
- Defining rule: equal turns multiply the distance from the centre by the same factor
- First described: Rene Descartes, in a letter of 1638
- Named marvellous by: Jacob Bernoulli, who died in 1705
- Golden spiral: the case that grows by about 1.618 each quarter turn
- Nautilus shell: grows by roughly 1.33 each quarter turn, so it is not golden
Often mistaken for something else: An Archimedean spiral keeps its coils the same distance apart; a logarithmic spiral spaces them wider and wider in a constant ratio.
The story behind Logarithmic Spiral
Rene Descartes described the curve in a letter of 1638. Jacob Bernoulli studied it late in the seventeenth century and named it spira mirabilis, the marvellous spiral, because it reappears unchanged under so many transformations. He asked for it to be cut on his tomb at Basel with a Latin motto meaning though changed, I rise again the same; the mason carved a different kind of spiral. In 1917 D'Arcy Thompson explained in On Growth and Form why shells and horns grow this way.
Why Logarithmic Spiral matters
Logarithmic Spiral separates a true fact, that growth without change of shape makes this curve, from the false one that every natural spiral is golden.
How to read Logarithmic Spiral
Mathematicians use the term for the whole family of curves; writers on sacred geometry usually mean the golden case and often call a nautilus one.
Logarithmic Spiral in practice: an example
A reader with a photograph of a cut nautilus shell and a ruler can measure the width of the coil at each quarter turn and see which spiral it is.
Where readings of Logarithmic Spiral go wrong
A spiral laid over a photograph of a shell, a storm or a face proves nothing until the growth rate is measured.
Sources and editorial method
This entry summarizes the tradition in the Almanac's own words; where no source is listed, it is an editorial summary rather than a primary text, and symbolic readings are named as the tradition's, not as fact.
- Source
- The Great Almanac Dictionary
- Category
- Mystical Systems
- Aliases
- golden spiral, equiangular spiral, logarithmic spiral in nature, nautilus spiral, spira mirabilis
- Last updated
- 2026-09-26
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Common questions
What is a logarithmic spiral?
It is a spiral whose distance from the centre grows by a constant factor for each equal angle of turn. It looks the same at every scale, which is why it describes things that grow while keeping their shape.
Is the nautilus shell a golden spiral?
No. It is a logarithmic spiral, but its growth rate is different. A golden spiral widens by about 1.618 each quarter turn, while measured nautilus shells widen by roughly 1.33.
What is the difference between a golden spiral and a Fibonacci spiral?
The golden spiral is an exact logarithmic spiral with a growth factor of the golden ratio per quarter turn. The Fibonacci spiral is drawn from quarter circles in squares of sides 1, 1, 2, 3, 5 and so on, and only approximates it.
