helixes

A helix is a smooth space curve obtained when a point moves around a fixed axis with constant angular velocity while also advancing parallel to the axis. This combination of constant radius about the axis and constant rise per turn produces a three‑dimensional spiral that is not confined to a plane. When the cross‑section around the axis is circular, the curve is called a circular helix; more general helices may have other cross‑sections or orientations of the axis.

In common parametrizations, a circular helix around the z‑axis is written as x = R cos t, y = R sin t, z = (p/2π) t, where R is the radius and p is the pitch, the vertical distance gained after one full turn (t increases by 2π). Equivalently, x = R cos t, y = R sin t, z = a t with a = p/(2π). The axis and direction define the handedness: right‑handed helices turn in a clockwise sense looking from the positive axis, left‑handed in the opposite sense.

Geometrically, circular helices have constant curvature κ and torsion τ, given by κ = R/(R² + a²) and τ = a/(R² + a²) for radius R and slope a. These invariants reflect the uniform winding of the curve.

Helices occur in nature and technology. The DNA double helix is a famous biological example, with a characteristic pitch and radius. In proteins, alpha helices are common structural motifs. Mechanical devices employ helical springs and threaded fasteners that rely on the geometry of helices to convert linear and rotational motion. Helical structures also appear in materials science and nanotechnology, including carbon nanotubes and other nanoscale coils.