Coiledspiral

Coiledspiral is a term used across mathematics, graphics, and design to describe a curve or pattern that blends a spiral with an additional coiling motion. It is not a single standard object, but a descriptive label for several related constructions where a spiral grows or wraps while undergoing a secondary winding.

One common 2D construction defines a parametric curve by x(θ) = (a + b θ) cos(m θ), y(θ) = (a + b θ) sin(m θ), with a > 0, b > 0, and m > 0. Here the radius increases linearly with θ (a + b θ) as the angle is scaled by m, creating a spiral that also coils around its center. The parameter m controls the density of the secondary winding; integer values yield whole-number wraps within a given θ-range.

A 3D variant models a coil that winds along a spiraling axis, for example X(t) = ((R + α t) cos t, (R + α t) sin t, β t). This produces a helical thread whose axis itself follows a planar spiral, producing a double-geared coil appearance.

Applications include decorative patterns in textile and graphic design, jewelry, architectural ornament, and algorithmic art. In mathematics, coiledspiral is used descriptively to discuss families of curves that interpolate between Archimedean or logarithmic spirals and various coiled surfaces.

Related concepts include the spiral, helix, Archimedean spiral, and logarithmic spiral. The term’s usage varies by field and is intended as a versatile descriptor rather than a fixed formal object.