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17
Construction of Triply Periodic Minimal Surfaces
, 1996
"... We discuss the construction of triply period minimal surfaces. This includes concepts for constructing new examples as well as a discussion of numerical computations based on the new concept of discrete minimal surfaces. As a result we present a wealth of old and new examples and suggest directio ..."
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Cited by 10 (2 self)
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We discuss the construction of triply period minimal surfaces. This includes concepts for constructing new examples as well as a discussion of numerical computations based on the new concept of discrete minimal surfaces. As a result we present a wealth of old and new examples and suggest directions for further generalizations.
A FAMILY OF TRIPLY PERIODIC COSTA SURFACES
 PACIFIC JOURNAL OF MATHEMATICS
, 2003
"... We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves. ..."
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Cited by 5 (1 self)
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We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.
Infinite Periodic Discrete Minimal Surfaces without SelfIntersections
 Balkan J. Geom. Appl
"... A triangulated piecewiselinear minimal surface in Euclidean 3space defined using a variational characterization is critical for area amongst all continuous piecewiselinear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfac ..."
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Cited by 3 (0 self)
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A triangulated piecewiselinear minimal surface in Euclidean 3space defined using a variational characterization is critical for area amongst all continuous piecewiselinear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfaces that are embedded and are periodic in three independent directions of .
Description of Cubic Liquidcrystalline Structures using Simple
"... In lyotropic and thermotropic liquid crystals of cubic symmetry the molecular organisation is naturally described in relation to a central partitioning surface, which is frequently assumed to be a minimal surface. Here we suggest a mathematically simpler basis, borrowed from solidstate physics, giv ..."
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Cited by 1 (0 self)
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In lyotropic and thermotropic liquid crystals of cubic symmetry the molecular organisation is naturally described in relation to a central partitioning surface, which is frequently assumed to be a minimal surface. Here we suggest a mathematically simpler basis, borrowed from solidstate physics, given by the leading invariants of the threedimensional Fourier series for the space group. The zero level surface is taken as the central partition, with the complete family of level surfaces then providing the foliation of space either side of it. In particular, we analyse the space group l mh and establish a topological classification of the simplest central partitions and their accompanying families of foliations. Further, we quantitatively compare the particular subclass which shares the topology of the familiar P minimal surface, and find that it yields extremely accurate approximations. Amongst the proposed applications of our analysis, we illustrate the construction of spacefilling director fields for 'blue ' phases. The selfassembly of surfactant (or surfactantlike lipid) molecules in solution can produce an incredible diversity of aggregate phases. Of the various lyotropic liquidcrystalline phases identified by Xray (or neutron) diffraction experiments, those of cubic symmetry have presented the greatest
Morphological characterizationof spatial patterns
"... We describe a morphological image analysis method to characterize spatial patterns in terms of geometry and topology. This involves the calculation of the Minkowski functionals and the analysis of these functionals as a function of control parameters. The method is applied to triply periodic minimal ..."
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We describe a morphological image analysis method to characterize spatial patterns in terms of geometry and topology. This involves the calculation of the Minkowski functionals and the analysis of these functionals as a function of control parameters. The method is applied to triply periodic minimal surfaces and to threedimensional structures formed in polymer solutions.
Symmetric Capillary Surfaces In A Cube  Part 3. More Exotic Surfaces, Gravity
"... Previous numerical experiments are extended to more complex surfaces and in some cases to nonzero gravity. 1. The Mathematical Problem Let \Phi be the unit cube in R 3 . We are considering subdomains\Omega ae \Phi having a piecewise smooth boundary @\Omega = \Gamma [ \Sigma, where \Gamma is a ..."
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Previous numerical experiments are extended to more complex surfaces and in some cases to nonzero gravity. 1. The Mathematical Problem Let \Phi be the unit cube in R 3 . We are considering subdomains\Omega ae \Phi having a piecewise smooth boundary @\Omega = \Gamma [ \Sigma, where \Gamma is a subset of the interior of \Phi and \Sigma is a subset of the boundary @\Phi of \Phi. We are looking for those subdomains \Omega which solve the following variational problem: The energy functional E = Z \Gamma d\Gamma + b Z \Omega x 3 d\Omega \Gamma cos # Z \Sigma d\Sigma is minimal under the restriction that the volume V = Z \Omega d\Omega attaines a prescribed value. It is a well known fact  going back to K. F. Gauß  that solutions of this variational problem must be such that the capillary surface \Gamma has prescribed mean curvature 2H = bx 3 + p and the contact angle between \Gamma and \Sigma is equal to # (see, e.g., [5]). The mean curvature is thus constant if the Bond n...
The {4, 5} isogonal sponges on the cubic lattice
"... Isogonal polyhedra are those polyhedra having the property of being vertextransitive. By this we mean that every vertex can be mapped to any other vertex via a symmetry of the whole polyhedron; in a sense, every vertex looks exactly like any other. The Platonic solids are examples, but these are bou ..."
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Isogonal polyhedra are those polyhedra having the property of being vertextransitive. By this we mean that every vertex can be mapped to any other vertex via a symmetry of the whole polyhedron; in a sense, every vertex looks exactly like any other. The Platonic solids are examples, but these are bounded polyhedra and our focus here is on infinite polyhedra. When the polygons of an infinite isogonal polyhedron are all planar and regular, the polyhedra are also known as sponges, pseudopolyhedra, or infinite skew polyhedra. These have been studied over the years, but many have been missed by previous researchers. We first introduce a notation for labeling threedimensional isogonal polyhedra and then show how this notation can be combinatorially used to find all of the isogonal polyhedra that can be created given a specific vertex star configuration. As an example, we apply our methods to the {4, 5} vertex star of five squares aligned along the planes of a cubic lattice and prove that there are exactly 15 such unlabeled sponges and 35 labeled ones. Previous efforts had found only 8 of the 15 shapes. 1
COPLANAR CONSTANT MEAN CURVATURE SURFACES
, 2005
"... Abstract. We consider constant mean curvature surfaces of finite topology, properly embedded in threespace in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an ..."
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Abstract. We consider constant mean curvature surfaces of finite topology, properly embedded in threespace in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane: we construct and classify the entire family of these coplanar constant mean curvature surfaces. Dedicated to Hermann Karcher on the occasion of his sixtyfifth birthday.
NEW COMPONENTS OF THE MODULI SPACE OF MINIMAL SURFACES IN 4DIMENSIONAL FLAT TORI
, 2004
"... Abstract. In this paper, we consider new components of a key space of a Moduli space of minimal surfaces in flat 4tori and calculate their dimensions. Moreover, we construct an example of minimal surfaces in 4tori and obtain an element of the Moduli. In the process of the construction, we give an ..."
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Abstract. In this paper, we consider new components of a key space of a Moduli space of minimal surfaces in flat 4tori and calculate their dimensions. Moreover, we construct an example of minimal surfaces in 4tori and obtain an element of the Moduli. In the process of the construction, we give an example of minimal surfaces with good property in a 3torus distinct from classical examples. 1.