Helicoid
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The helicoid is one of the first minimal surfaces discovered. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through that point.
The helicoid is also a ruled surface, meaning that it is a trace of a line. Alternatively, for any point on the surface, there is a line on the surface passing through it.
The helicoid and the catenoid are parts of a family of helicoid-catenoid minimal surfaces.
The helicoid is shaped like the Archimedes' screw, but extends infinitely in all directions. It can be described by the following parametric equations in Cartesian coordinates:
- <math> x = \rho \cos \theta, \ <math>
- <math> y = \rho \sin \theta, \ <math>
- <math> z = \alpha \theta, \ <math>
where both ρ and θ range from negative infinity to positive infinity.
The helicoid is homeomorphic to the plane <math> \mathbb{R}^2 <math>. To see this, let alpha decrease continuously from its given value down to zero. Each intermediate value of α will describe a different helicoid, until α = 0 is reached and the helicoid becomes a plane (a plane is a degenerate helicoid).
A plane can be turned into a helicoid by choosing a line on the plane (call it an axis) then twisting the plane around that axis.