Helical surface

Section of a helical surface for

{\ displaystyle 0 \ leq r \ leq r_ {0}, 0 \ leq \ phi \ leq \ phi _ {0}}

The helical surface or helicoid is a surface from the mathematical sub-area of differential geometry . Besides the plane, it is the only simply connected minimal surface in 3-dimensional Euclidean space .

Parameterization

A section of the helical surface for the parameter . The excerpt shows the part for and .

{\ displaystyle c = 1}

{\ displaystyle -1 \ leq r \ leq 1}

{\ displaystyle - \ pi \ leq \ phi \ leq \ pi}

The helical surface is parameterized for a fixed constant ${\ displaystyle c> 0}$

{\ displaystyle x = r \ cos (\ phi)}

{\ displaystyle y = r \ sin (\ phi)}

{\ displaystyle z = c \ phi}

,

where and assume all real values, i.e. run from to . ${\ displaystyle r}$ ${\ displaystyle \ phi}$ ${\ displaystyle - \ infty}$ ${\ displaystyle \ infty}$

Minimal area

The main curvatures of the helical surface in the point corresponding to the parameters are and , the mean curvature is therefore zero in every point, the helical surface is a minimum surface . ${\ displaystyle r, \ phi}$ ${\ displaystyle {\ frac {1} {1 + r ^ {2}}}}$ ${\ displaystyle - {\ frac {1} {1 + r ^ {2}}}}$

Topologically it is homeomorphic to the plane.

Locally it is isometric to the catenoid , but it is not homeomorphic to the latter.

It is a ruled surface and a screw surface . It can also be represented as a sliding surface .

Practical and Scientific Meaning - Chirality

Chemistry: ( P ) - (+) - Heptahelicene exhibits clockwise helicity .
Chemistry: ( M ) - (-) - heptahelicene exhibits counterclockwise helicity.
Spiral staircase in the Vatican Museum in Rome as an example of a helical structure for comparison. The symmetry corresponds to that of ( P ) -Heptahelicen
The screw rotation of this climbing plant corresponds to the direction of rotation of ( M ) -Heptahelicen

There are numerous areas of application for helical surfaces in nature, in architecture and in chemistry. The direction of rotation (chirality) also plays a role.

history

The helicoid was described by Euler and Meusnier in the 18th century . Catalan proved in 1842 that it is the only minimal ruled surface besides the plane. Meeks and Rosenberg proved in 2005 (based on the inequalities of Colding-Minicozzi) that there are only 2 types of simply connected minimal surfaces in : the plane and the helicoids. For non-zero topological gender, David Allen Hoffman and colleagues in the 1990s found further examples that emerged from the Helicoide. The proof that they form a completely embeddable minimal surface for genus 1 was provided by Hoffman, Michael Wolf and Matthias Weber in 2009 (before that, except for the case of sex 0, this was only proven for the case of infinite sex). ${\ displaystyle \ mathbb {R} ^ {3}}$

Individual evidence

^ William H. Meeks, Harold Rosenberg (2005). The uniqueness of the helicoid. Annals of Mathematics (2), 161 (2), 727-758 doi : 10.4007 / annals.2005.161.727
↑ Tobias H. Colding, William P. Minicozzi (2004). The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected. Annals of Mathematics (2), 160 (2), 573-615 doi : 10.4007 / annals.2004.160.573
↑ David Allen Hoffman, Matthias Weber, Michael Wolf: An embedded genus-one helicoid, Annals of Mathematics, Volume 169, 2009, pp. 347-448 (and Proc. Nat. Acad. USA, Volume 102, 2005, p. 16566 -16568)

Web links

Commons : Helicoids - collection of images, videos and audio files

Helicoid : Collection of Images and Animations (Matthias Weber, Indiana University)

[1] William H. Meeks, Harold Rosenberg (2005). The uniqueness of the helicoid. Annals of Mathematics (2), 161 (2), 727-758 doi : 10.4007 / annals.2005.161.727

[2] Tobias H. Colding, William P. Minicozzi (2004). The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected. Annals of Mathematics (2), 160 (2), 573-615 doi : 10.4007 / annals.2004.160.573

[3] David Allen Hoffman, Matthias Weber, Michael Wolf: An embedded genus-one helicoid, Annals of Mathematics, Volume 169, 2009, pp. 347-448 (and Proc. Nat. Acad. USA, Volume 102, 2005, p. 16566 -16568)