Minimal Surfaces for Architectural Constructions Udc 72.01(083.74)(045)=111
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Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia
Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia Brief information on some classes of the surfaces which cylinders, cones and ortoid ruled surfaces with a constant were not picked out into the special section in the encyclo- distribution parameter possess this property. Other properties pedia is presented at the part “Surfaces”, where rather known of these surfaces are considered as well. groups of the surfaces are given. It is known, that the Plücker conoid carries two-para- At this section, the less known surfaces are noted. For metrical family of ellipses. The straight lines, perpendicular some reason or other, the authors could not look through to the planes of these ellipses and passing through their some primary sources and that is why these surfaces were centers, form the right congruence which is an algebraic not included in the basic contents of the encyclopedia. In the congruence of the4th order of the 2nd class. This congru- basis contents of the book, the authors did not include the ence attracted attention of D. Palman [8] who studied its surfaces that are very interesting with mathematical point of properties. Taking into account, that on the Plücker conoid, view but having pure cognitive interest and imagined with ∞2 of conic cross-sections are disposed, O. Bottema [9] difficultly in real engineering and architectural structures. examined the congruence of the normals to the planes of Non-orientable surfaces may be represented as kinematics these conic cross-sections passed through their centers and surfaces with ruled or curvilinear generatrixes and may be prescribed a number of the properties of a congruence of given on a picture. -
On Algebraic Minimal Surfaces
On algebraic minimal surfaces Boris Odehnal December 21, 2016 Abstract 1 Introduction Minimal surfaces have been studied from many different points of view. Boundary We give an overiew on various constructions value problems, uniqueness results, stabil- of algebraic minimal surfaces in Euclidean ity, and topological problems related to three-space. Especially low degree exam- minimal surfaces have been and are stil top- ples shall be studied. For that purpose, we ics for investigations. There are only a use the different representations given by few results on algebraic minimal surfaces. Weierstraß including the so-called Björ- Most of them were published in the second ling formula. An old result by Lie dealing half of the nine-teenth century, i.e., more with the evolutes of space curves can also or less in the beginning of modern differen- be used to construct minimal surfaces with tial geometry. Only a few publications by rational parametrizations. We describe a Lie [30] and Weierstraß [50] give gen- one-parameter family of rational minimal eral results on the generation and the prop- surfaces which touch orthogonal hyperbolic erties of algebraic minimal surfaces. This paraboloids along their curves of constant may be due to the fact that computer al- Gaussian curvature. Furthermore, we find gebra systems were not available and clas- a new class of algebraic and even rationally sical algebraic geometry gained less atten- parametrizable minimal surfaces and call tion at that time. Many of the compu- them cycloidal minimal surfaces. tations are hard work even nowadays and synthetic reasoning is somewhat uncertain. Keywords: minimal surface, algebraic Besides some general work on minimal sur- surface, rational parametrization, poly- faces like [5, 8, 43, 44], there were some iso- nomial parametrization, meromorphic lated results on algebraic minimal surfaces function, isotropic curve, Weierstraß- concerned with special tasks: minimal sur- representation, Björling formula, evolute of faces on certain scrolls [22, 35, 47, 49, 53], a spacecurve, curve of constant slope. -
Riemannian Submanifolds: a Survey
RIEMANNIAN SUBMANIFOLDS: A SURVEY BANG-YEN CHEN Contents Chapter 1. Introduction .............................. ...................6 Chapter 2. Nash’s embedding theorem and some related results .........9 2.1. Cartan-Janet’s theorem .......................... ...............10 2.2. Nash’s embedding theorem ......................... .............11 2.3. Isometric immersions with the smallest possible codimension . 8 2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature .......................................... ..................12 2.5. Isometric immersions with prescribed mean curvature. ...........13 Chapter 3. Fundamental theorems, basic notions and results ...........14 3.1. Fundamental equations ........................... ..............14 3.2. Fundamental theorems ............................ ..............15 3.3. Basic notions ................................... ................16 3.4. A general inequality ............................. ...............17 3.5. Product immersions .............................. .............. 19 3.6. A relationship between k-Ricci tensor and shape operator . 20 3.7. Completeness of curvature surfaces . ..............22 Chapter 4. Rigidity and reduction theorems . ..............24 4.1. Rigidity ....................................... .................24 4.2. A reduction theorem .............................. ..............25 Chapter 5. Minimal submanifolds ....................... ...............26 arXiv:1307.1875v1 [math.DG] 7 Jul 2013 5.1. First and second variational formulas -
Deformations of the Gyroid and Lidinoid Minimal Surfaces
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES ADAM G. WEYHAUPT Abstract. The gyroid and Lidinoid are triply periodic minimal surfaces of genus 3 embed- ded in R3 that contain no straight lines or planar symmetry curves. They are the unique embedded members of the associate families of the Schwarz P and H surfaces. In this paper, we prove the existence of two 1-parameter families of embedded triply periodic minimal surfaces of genus 3 that contain the gyroid and a single 1-parameter family that contains the Lidinoid. We accomplish this by using the flat structures induced by the holomorphic 1 1-forms Gdh, G dh, and dh. An explicit parametrization of the gyroid using theta functions enables us to find a curve of solutions in a two-dimensional moduli space of flat structures by means of an intermediate value argument. Contents 1. Introduction 2 2. Preliminaries 3 2.1. Parametrizing minimal surfaces 3 2.2. Cone metrics 5 2.3. Conformal quotients of triply periodic minimal surfaces 5 3. Parametrization of the gyroid and description of the periods 7 3.1. The P Surface and tP deformation 7 3.2. The period problem for the P surface 11 3.3. The gyroid 14 4. Proof of main theorem 16 4.1. Sketch of the proof 16 4.2. Horizontal and vertical moduli spaces for the tG family 17 4.3. Proof of the tG family 20 5. The rG and rL families 25 5.1. Description of the Lidinoid 25 5.2. Moduli spaces for the rL family 29 5.3. -
A Chiral Family of Triply-Periodic Minimal Surfaces from the Quartz Network
A chiral family of triply-periodic minimal surfaces from the quartz network Shashank G. Markande 1, Matthias Saba2, G. E. Schroder-Turk¨ 3 and Elisabetta A. Matsumoto 1∗ 1School of Physics, Georgia Tech, Atlanta, GA 30318, USA 2 Department of Physics, Imperial College, London, UK 3School of Engineering and IT, Mathematics and Statistics, Murdoch University, Murdoch, Australia ∗ Author for correspondence: [email protected] May 21, 2018 Abstract We describe a new family of triply-periodic minimal surfaces with hexagonal symmetry, related to the quartz (qtz) and its dual (the qzd net). We provide a solution to the period problem and provide a parametrisation of these surfaces, that are not in the regular class, by the Weierstrass-Enneper formalism. We identified this analytical description of the surface by generating an area-minimising mesh interface from a pair of dual graphs (qtz & qzd) using the generalised Voronoi construction of De Campo, Hyde and colleagues, followed by numerical identification of the flat point structure. This mechanism is not restricted to the specific pair of dual graphs, and should be applicable to a broader set of possible dual graph topologies and their corresponding minimal surfaces, if existent. 1 Introduction The discovery of cubic structures with vivid photonic properties in a wide range of biological systems, such as those responsible for the structural colouring seen in the wing scales of butterflies [1, 2, 3, 4] and in beetle cuticles [5, 6], has driven a renewed interest in mathematics of minimal surfaces – surfaces which, every- where, have zero mean curvature. The optical properties in these structures comes from the periodic nature of their bulk structure. -
Blackfolds, Plane Waves and Minimal Surfaces
Blackfolds, Plane Waves and Minimal Surfaces Jay Armas1;2 and Matthias Blau2 1 Physique Th´eoriqueet Math´ematique Universit´eLibre de Bruxelles and International Solvay Institutes ULB-Campus Plaine CP231, B-1050 Brussels, Belgium 2 Albert Einstein Center for Fundamental Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland [email protected], [email protected] Abstract Minimal surfaces in Euclidean space provide examples of possible non-compact horizon ge- ometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configura- tions compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter space-times in which case minimal surfaces can be static and compact. We use the blackfold approach in order to scan for possible black hole horizon geometries and topologies in asymptotically flat, plane wave and de Sitter space-times. In the process we uncover several new configurations, such as black arXiv:1503.08834v2 [hep-th] 6 Aug 2015 helicoids and catenoids, some of which have an asymptotically flat counterpart. In particular, we find that the ultraspinning regime of singly-spinning Myers-Perry black holes, described in terms of the simplest minimal surface (the plane), can be obtained as a limit of a black helicoid, suggesting that these two families of black holes are connected. We also show that minimal surfaces embedded in spheres rather than Euclidean space can be used to construct static compact horizons in asymptotically de Sitter space-times. -
Classical Minimal Surfaces in Euclidean Space by Examples
Classical Minimal Surfaces in Euclidean Space by Examples Geometric and Computational Aspects of the Weierstra¼ representation September 25, 2001 Matthias Weber 2 Contents 1 Introduction 6 2 Basic Di®erential Geometry Of Surfaces 7 2.1 Local Surface Parameterizations . 7 2.2 The Stereographic Projection . 8 2.3 The Normal Vector or Gau¼ Map . 9 3 The Weierstrass Representation 11 3.1 The Gau¼ map . 11 3.2 Harmonic coordinate functions . 12 3.3 Weierstrass data . 13 3.4 The ¯rst examples . 15 3.5 Local analysis at regular points . 17 3.6 The associate family . 19 3.7 The conjugate surface and symmetries . 20 4 Minimal surfaces on Punctured Spheres 22 4.1 Period conditions . 22 4.2 A surface with a planar end and an Enneper end . 23 4.3 A sphere with one planar and two catenoid ends . 24 4.4 The k-Noids of Jorge and Meeks . 25 5 The Scherk Surfaces 28 5.1 The singly periodic Scherk surface . 28 3 5.2 The period conditions . 28 5.3 Changing the angle between the ends . 30 5.4 The doubly periodic Scherk surface . 32 5.5 Singly periodic Scherk surfaces with higher dihedral symmetry 34 5.6 The twist deformation of Scherk's singly periodic surface . 35 5.7 The Schwarz-Christo®el formula . 39 6 Minimal Surfaces De¯ned On Punctured Tori 41 6.1 Complex tori . 41 6.2 Algebraic equations . 42 6.3 Theta functions . 43 6.4 Elliptic functions via Schwarz-Christo®el maps . 44 6.5 The Chen-Gackstatter surface . -
What Are Minimal Surfaces?
What are Minimal surfaces? RUKMINI DEY International Center for Theoretical Sciences (ICTS-TIFR), Bangalore ICTS-TIFR, Bangalore May, 2018 Topology of compact oriented surfaces – genus Sphere, torus, double torus are usual names you may have heard. Sphere and ellipsoids are abundant in nature – for example a ball and an egg. They are topologically “same” – in the sense if they were made out of plasticine, you could have deformed one to the other. Figure 1: Sphere and a glass ball Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 2 of 46 Figure 2: Ellipsoids and an egg Figure 3: Tori and donut Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 3 of 46 Figure 4: Triple torus and surface of a pretzel Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 4 of 46 Handle addition to the sphere –genus Figure 5: Triple torus In fact as the above figure suggests one can construct more and more complicated sur- faces by adding handles to a sphere – the number of handles added is the genus. Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 5 of 46 Genus classifies compact oriented surfaces topologically Sphere (genus g =0) is not topologically equivalent to torus (genus g =1) and none of them are topologically equivalent to double torus (genus g =2) and so on. One cannot continuously deform one into the other if the genus-es are different. F Euler’s formula: Divide the surface up into faces (without holes) with a certain number of vertices and edges. Let no. of vertices = V , no. of faces = F , no. of edges = E, no. -
Stratified Multivarifolds, and the Plateau Problem
Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem BA0 TRONG THI A. T. FOMENKO TRANSLATIONS OF MATHEMATICAL MONOGRAPHS TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME84 Minimal Surfaces, Stratified M u ltiva rifo lds, and the Plateau Problem 8AO TRONG THI A. T. FOMENKO American Mathematical Society ProvidenceRhode island AAO 'IOHI' TXH A. T. 4)OMEHKO MHHHMAJIbHbIE IIOBEPXHOCTH H HPOEJIEMA HJIATO «HAYKA», MOCKBA, 1987 Translated from the Russian by E. J. F. Primrose Translation edited by Ben Silver 1980 Mathematics Subject Classification (1985 Revision). Primary49F10, 53A10; Secondary 58E12, 58E20. ABSTRACT. The book is an account of the current state of the theory of minimal surfaces and of one of the most important chapters of this theory-the Plateau problem, i.e. the problem of the existence of a minimal surface with boundary prescribed in advance. The authors exhibit deep connections of minimal surface theory with differential equations, Lie groups and Lie algebras, topology, and multidimensional variational calculus. The presentation is simplified to a large extent; the book is furnished with a wealth of illustrative material. Bibliography: 471 titles. 153 figures, 2 tables. Library of Congress Cataloging-ln-Publicatlon Data Dao, Trong Thi. [Minimal'nye poverkhnosti i problema Plato. English] Minimal surfaces, stratified multivarifolds, and the Plateau problem/Dio Trong Thi, A. T. Fomenko. p.cm. - (Translations of mathematical monographs; v. 84) Translation of: Minimal'nye poverkhnosti i problema Plato. Includes bibliographical references and index. ISBN 0-8218-4536-5 1. Surfaces, Minimal.2. Plateau's problem.I. Fomenko, A. T.II. Title.III. Series. QA644.D2813 1991 90-22932 516.3'62-dc2O CIP Copyright ©1991 by the American Mathematical Society. -
New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space
Fontbonne University GriffinShare All Theses, Dissertations, and Capstone Projects Theses, Dissertations, and Capstone Projects 2006 New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space Adam G. Weyhaupt Indiana University Follow this and additional works at: https://griffinshare.fontbonne.edu/all-etds Part of the Mathematics Commons This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Recommended Citation Weyhaupt, Adam G., "New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space" (2006). All Theses, Dissertations, and Capstone Projects. 160. https://griffinshare.fontbonne.edu/all-etds/160 NEW FAMILIES OF EMBEDDED TRIPLY PERIODIC MINIMAL SURFACES OF GENUS THREE IN EUCLIDEAN SPACE Adam G. Weyhaupt Submitted to the faculty of the University Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Department of Mathematics Indiana University August 2006 Accepted by the Graduate Faculty, Indiana University, in partial fulfillment of the require- ments for the degree of Doctor of Philosophy. Matthias Weber, Ph.D. Jiri Dadok, Ph.D. Bruce Solomon, Ph.D. Peter Sternberg, Ph.D. 4 August 2006 ii Copyright 2006 Adam G. Weyhaupt ALL RIGHTS RESERVED iii To Julia — my source of strength, my constant companion, my love forever and to Brandon and Ryan — whose exuberance and smiles lift my spirits daily and have changed my life in so many wonderful ways. iv Acknowledgements I can not imagine having completed graduate school with a different advisor: thank you, Matthias Weber, for your encouragement, your guidance, your constant availability, and for making me feel like a colleague. -
What Are Minimal Surfaces?
What are minimal surfaces? RUKMINI DEY International Center for Theoretical Sciences (ICTS-TIFR), Bangalore SWSM, ICTS, Bangalore May , 2019 Topology of compact oriented surfaces – genus Sphere, torus, double torus are usual names you may have heard. Sphere and ellipsoids are abundant in nature – for example a ball and an egg. They are topologically “same” – in the sense if they were made out of plasticine, you could have deformed one to the other. Figure 1: Sphere and a glass ball Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 2 of 54 Figure 2: Ellipsoids and an egg Figure 3: Tori and donut Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 3 of 54 Figure 4: Triple torus and surface of a pretzel Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 4 of 54 Handle addition to the sphere –genus Figure 5: Triple torus In fact as the above figure suggests one can construct more and more complicated sur- faces by adding handles to a sphere – the number of handles added is the genus. Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 5 of 54 Genus classifies compact oriented surfaces topologically Sphere (genus g =0) is not topologically equivalent to torus (genus g =1) and none of them are topologically equivalent to double torus (genus g =2) and so on. One cannot continuously deform one into the other if the genus-es are different. F Euler’s formula: Divide the surface up into faces (without holes) with a certain number of vertices and edges. Let no. of vertices = V , no. of faces = F , no. of edges = E, no. -
Collected Atos
Mathematical Documentation of the objects realized in the visualization program 3D-XplorMath Select the Table Of Contents (TOC) of the desired kind of objects: Table Of Contents of Planar Curves Table Of Contents of Space Curves Surface Organisation Go To Platonics Table of Contents of Conformal Maps Table Of Contents of Fractals ODEs Table Of Contents of Lattice Models Table Of Contents of Soliton Traveling Waves Shepard Tones Homepage of 3D-XPlorMath (3DXM): http://3d-xplormath.org/ Tutorial movies for using 3DXM: http://3d-xplormath.org/Movies/index.html Version November 29, 2020 The Surfaces Are Organized According To their Construction Surfaces may appear under several headings: The Catenoid is an explicitly parametrized, minimal sur- face of revolution. Go To Page 1 Curvature Properties of Surfaces Surfaces of Revolution The Unduloid, a Surface of Constant Mean Curvature Sphere, with Stereographic and Archimedes' Projections TOC of Explicitly Parametrized and Implicit Surfaces Menu of Nonorientable Surfaces in previous collection Menu of Implicit Surfaces in previous collection TOC of Spherical Surfaces (K = 1) TOC of Pseudospherical Surfaces (K = −1) TOC of Minimal Surfaces (H = 0) Ward Solitons Anand-Ward Solitons Voxel Clouds of Electron Densities of Hydrogen Go To Page 1 Planar Curves Go To Page 1 (Click the Names) Circle Ellipse Parabola Hyperbola Conic Sections Kepler Orbits, explaining 1=r-Potential Nephroid of Freeth Sine Curve Pendulum ODE Function Lissajous Plane Curve Catenary Convex Curves from Support Function Tractrix