Helicoid

# Helicoid

Overview
The helicoid, after the plane and the catenoid
Catenoid
A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...

, is the third minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....

to be known. It was first discovered by Jean Baptiste Meusnier
Jean Baptiste Meusnier
Jean Baptiste Marie Charles Meusnier de la Place was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature of surfaces, which he formulated while he was at the École Royale du Génie . He also discovered the helicoid...

in 1776. Its name
Nomenclature
Nomenclature is a term that applies to either a list of names or terms, or to the system of principles, procedures and terms related to naming - which is the assigning of a word or phrase to a particular object or property...

derives from its similarity to the helix
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

: for every point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

on the helicoid there is a helix contained in the helicoid which passes through that point. Since it is considered that the planar range extends through negative and positive infinity, close observation shows the appearance of two parallel or mirror planes in the sense that if the slope of one plane is traced, the co-plane can be seen to be bypassed or skipped, though in actuality the co-plane is also traced from the opposite perspective.
Discussion

Encyclopedia
The helicoid, after the plane and the catenoid
Catenoid
A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...

, is the third minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....

to be known. It was first discovered by Jean Baptiste Meusnier
Jean Baptiste Meusnier
Jean Baptiste Marie Charles Meusnier de la Place was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature of surfaces, which he formulated while he was at the École Royale du Génie . He also discovered the helicoid...

in 1776. Its name
Nomenclature
Nomenclature is a term that applies to either a list of names or terms, or to the system of principles, procedures and terms related to naming - which is the assigning of a word or phrase to a particular object or property...

derives from its similarity to the helix
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

: for every point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

on the helicoid there is a helix contained in the helicoid which passes through that point. Since it is considered that the planar range extends through negative and positive infinity, close observation shows the appearance of two parallel or mirror planes in the sense that if the slope of one plane is traced, the co-plane can be seen to be bypassed or skipped, though in actuality the co-plane is also traced from the opposite perspective.

The helicoid is also a ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...

(and a right conoid
Right conoid
In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly a fixed straight line, called the axis of the right conoid....

), 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. Indeed, Catalan
Eugène Charles Catalan
Eugène Charles Catalan was a French and Belgian mathematician.- Biography :Catalan was born in Bruges , the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville...

proved in 1842 that the helicoid and the plane were the only ruled minimal surfaces.

The helicoid and the catenoid
Catenoid
A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...

are parts of a family of helicoid-catenoid minimal surfaces.

The helicoid is shaped like Archimedes' screw
Archimedes' screw
The Archimedes' screw, also called the Archimedean screw or screwpump, is a machine historically used for transferring water from a low-lying body of water into irrigation ditches...

, but extends infinitely in all directions. It can be described by the following parametric equation
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

s in Cartesian coordinates:
where ρ and θ range from negative infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

to positive infinity, while α is a constant. If α is positive then the helicoid is right-handed as shown in the figure; if negative then left-handed.

The helicoid has principal curvature
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.-Discussion:...

s . The sum of these quantities gives the mean curvature
Mean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....

(zero since the helicoid is a minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....

) and the product gives the Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

.

The helicoid is homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

to the plane . To see this, let alpha decrease continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

ly from its given value down to zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

. Each intermediate value of α will describe a different helicoid, until α = 0 is reached and the helicoid becomes a vertical plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

.

Conversely, a plane can be turned into a helicoid by choosing a line, or axis, on the plane then twisting the plane around that axis.

For example, if we take h as the maxium value at z and R the radium, the area of the surface is π[R√(R²+h²)+h2*ln(R +√(R²+h²)/h)]

## Helicoid and catenoid

The helicoid and the catenoid
Catenoid
A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...

are locally isometric surfaces, see discussion there.