Spiral Types Compared

How to read a polar equation

A spiral is easiest to describe in polar coordinates. θ says which direction to look from the center; r says how far to travel in that direction. As θ increases, the rule for r determines how tightly the curve winds.

Plot the same increase in θ again and again. If r adds a constant, turns stay evenly spaced. If r multiplies by a constant, the gaps expand. If r grows with √θ, the curve packs area evenly.

Archimedean: equal spacing

In an Archimedean spiral, radius grows linearly with angle. Every complete revolution adds the same amount to r, so neighboring turns remain a constant distance apart.

That predictable spacing makes the curve useful for grooves, coils, rolled materials, and scanning paths. An old vinyl record approximates a very tight Archimedean spiral because each revolution must sit beside the last.

Fermat: equal area

A Fermat spiral grows with the square root of angle. Its turns get closer together as it moves outward, but points sampled at equal angle intervals occupy roughly equal areas.

The phyllotaxis point pattern is a discrete cousin of this spiral. Draw points with r = c√n and θ = nα, then connect points that appear to be neighbors: many Fermat-like arms become visible.

Logarithmic: the self-similar spiral

A logarithmic spiral grows exponentially. Rotate it by any angle and the new curve is a scaled copy of the old one. Its angle of intersection with radial lines never changes, which is why it is also called an equiangular spiral.

This is the family associated with idealized nautilus shells, cyclone bands, and some galaxies, although real natural forms only approximate the equation. A golden spiral is one specific logarithmic spiral whose radius grows by φ every quarter turn.

Which spiral should you use?

Choose by behavior, not appearance. For uniform track spacing, use Archimedean. For even point density, use Fermat. For scale-invariant growth, use logarithmic.

A drawing cropped to one turn can hide the distinction. Always compare several revolutions and measure how the gaps change.

• ✳Archimedean — coils, grooves, scanning paths
• ✳Fermat — phyllotaxis and even-area sampling
• ✳Logarithmic — self-similar growth and equiangular paths