Minimal surfaces with self-intersections along straight lines. I. Derivation and properties

1999 ◽  
Vol 55 (1) ◽  
pp. 58-64 ◽  
Author(s):  
E. Koch ◽  
W. Fischer
Keyword(s):  
Minimal Surface ◽  
Symmetry Group ◽  
Minimal Surfaces ◽  
Special Kind ◽  
Branch Points ◽  
Euler Characteristics ◽  
Oriented Surface ◽  
Periodic Minimal Surface ◽  
Full Symmetry ◽  
Straight Lines

A special kind of three-periodic minimal surface has been studied, namely surfaces that are generated from disc-like-spanned skew polygons and that intersect themselves exclusively along straight lines. A new procedure for their derivation is introduced in this paper. Several properties of each such surface may be deduced from its generating polygon: the full symmetry group of the surface, its orientability, the symmetry group of the oriented surface, the pattern of self-intersections, the branch points of the surface, the symmetry and periodicity of the spatial subunits demarcated by the surface, and the Euler characteristics both of the surface and of the spatial subunits. The corresponding procedures are described and illustrated by examples.

Description of a 3-periodic minimal surface family with trigonal symmetry

1994 ◽  
Vol 209 (1) ◽  
Author(s):  
A. Fogden
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Systematic Analysis ◽  
Trigonal Symmetry ◽  
Triply Periodic Minimal Surfaces ◽  
Periodic Minimal Surface ◽  
Triply Periodic ◽  
The Family

AbstractA systematic analysis of a family of triply periodic minimal surfaces of genus seven and trigonal symmetry is given. The family is found to contain five such surfaces free from self-intersections, three of which are previously unknown. Exact parametrisations of all surfaces are provided using the Weierstrass representation.

scholarly journals Minimal Surfaces with Only Horizontal Symmetries

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Márcio Fabiano da Silva ◽  
Guillermo Antonio Lobos ◽  
Valério Ramos Batista
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Reflection Principle ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic ◽  
Horizontal Type ◽  
Straight Lines ◽  
Schwarz Reflection Principle

The Schwarz reflection principle states that a minimal surface S in ℝ3 is invariant under reflections in the plane of its principal geodesics and also invariant under 180°-rotations about its straight lines. We find new examples of embedded triply periodic minimal surfaces for which such symmetries are all of horizontal type.

Polar varieties and bipolar surfaces of minimal surfaces in the n-sphere

2021 ◽  
Author(s):  
Katsuhiro Moriya
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Minimal Immersion ◽  
The Other ◽  
Branch Points

AbstractFor a given minimal surface in the n-sphere, two ways to construct a minimal surface in the m-sphere are given. One way constructs a minimal immersion. The other way constructs a minimal immersion which may have branch points. The branch points occur exactly at each point where the original minimal surface is geodesic. If a minimal surface in the 3-sphere is given, then these ways construct Lawson’s polar variety and bipolar surface.

PROTEIN NANODOMAIN PATTERNS IN LIPIDIC BICONTINUOUS CUBIC PHASES

2002 ◽  
Vol 16 (07) ◽  
pp. 225-230 ◽  
Author(s):  
BORISLAV ANGELOV
Keyword(s):  
Membrane Proteins ◽  
Minimal Surface ◽  
Self Assembly ◽  
Minimal Surfaces ◽  
Unit Cell ◽  
The Self ◽  
Periodic Minimal Surface ◽  
Cubic Phases ◽  
Bicontinuous Cubic ◽  
Diamond Type

Ordered nanopatterns that match to the {6, 4} tiling of the diamond type infinite periodic minimal surface are generated. The construction of the unit cell, generally recognized as a Monkey Saddle, is done numerically using the exact Weierstrass–Enneper representation of minimal surfaces. The obtained patterns are a good model for the self-assembly nanodomain organizations of membrane proteins, compatible with 3- or 6-fold symmetries and formed upon reconstitution in bicontinuous cubic lipid phases.

Symmetry Reduction, Exact Solutions, and Conservation Laws of the (2+1)-Dimensional Dispersive Long Wave Equations

2009 ◽  
Vol 64 (9-10) ◽  
pp. 597-603 ◽  
Author(s):  
Zhong Zhou Dong ◽  
Yong Chen
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Symmetry Group ◽  
Infinite Number ◽  
Wave Equations ◽  
Direct Method ◽  
Discrete Groups ◽  
Point Symmetry Group ◽  
Long Wave ◽  
Full Symmetry

By means of the generalized direct method, we investigate the (2+1)-dimensional dispersive long wave equations. A relationship is constructed between the new solutions and the old ones and we obtain the full symmetry group of the (2+1)-dimensional dispersive long wave equations, which includes the Lie point symmetry group S and the discrete groups D. Some new forms of solutions are obtained by selecting the form of the arbitrary functions, based on their relationship. We also find an infinite number of conservation laws of the (2+1)-dimensional dispersive long wave equations.

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