Helicoid

The helicoid is a ruled minimal surface formed by a straight line that moves with constant angular velocity around a fixed axis while translating along the axis. In Euclidean three-space it can be described parametrically by X(u, v) = (u cos v, u sin v, k v), where u and v range over the real numbers and k > 0 is the pitch parameter. For each fixed v, the set {X(u, v) : u ∈ R} is a straight line, and as v increases the line rotates about the z-axis while the surface ascends.

As a mathematical object, the helicoid is a minimal surface, meaning its mean curvature is zero everywhere. It is not a surface of revolution, but it possesses screw symmetry: the surface is invariant under a combined rotation about the axis by an angle φ and a translation along the axis by k φ. The Gaussian curvature is negative at all nonzero points, and the surface is non-compact and simply connected.

Historically, the helicoid was studied in the 18th century by Euler. In the theory of minimal surfaces it is part of the associate family that also includes the catenoid; within this family, continuous deformations preserve minimality and can transform between the helicoid and the catenoid.

The helicoid serves as a canonical example of a non-planar ruled minimal surface and appears in mathematical illustrations, as well as in architectural and design contexts that exploit screw-like symmetry and aesthetic geometry.