Helicals

Helicals refer to one or more helices, three-dimensional curves that wind around a central axis. In mathematics and the natural and engineered world, a helix is defined by a steady advance along an axis combined with a rotation about that axis, producing a smooth, screw-like shape.

For the standard circular helix, the path can be written parametrically as x = R cos t, y = R sin t, z = (P/2π) t, where R is the radius and P is the pitch, the axial distance advanced per full turn. The pitch angle φ satisfies tan φ = P/(2πR). The length of one turn equals sqrt((2πR)^2 + P^2). Variants include helices with noncircular cross sections or variable radius, described by continuous combinations of rotation and translation along an axis.

Helicals occur widely in nature and technology. The DNA molecule forms a double helix; many proteins contain alpha helices; bacterial flagella have helical propellers; and coil springs or screw threads are mechanical realizations of a helix. Helical motifs also appear in plant tendrils, cables wound around poles, and in metamaterials designed to exhibit chiral or directional properties.

Terminology and related concepts include the general notion of a helix as a screw-like curve, the helicoid as a related surface generated by a line that rotates while translating, and the idea of a helix angle, which measures the tilt of the curve relative to the axis. The plural helicals is simply the plural of helix or of the term used to describe multiple helical structures.