A Surface F : M → R Is Minimal If: M Has MEAN CURVATURE = 0. Small
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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. -
DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE
DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 Curvature—Overview • Intuitively, describes “how much a shape bends” – Extrinsic: how quickly does the tangent plane/normal change? – Intrinsic: how much do quantities differ from flat case? N T B Curvature—Overview • Driving force behind wide variety of physical phenomena – Objects want to reduce—or restore—their curvature – Even space and time are driven by curvature… Curvature—Overview • Gives a coordinate-invariant description of shape – fundamental theorems of plane curves, space curves, surfaces, … • Amazing fact: curvature gives you information about global topology! – “local-global theorems”: turning number, Gauss-Bonnet, … Curvature—Overview • Geometric algorithms: shape analysis, local descriptors, smoothing, … • Numerical simulation: elastic rods/shells, surface tension, … • Image processing algorithms: denoising, feature/contour detection, … Thürey et al 2010 Gaser et al Kass et al 1987 Grinspun et al 2003 Curvature of Curves Review: Curvature of a Plane Curve • Informally, curvature describes “how much a curve bends” • More formally, the curvature of an arc-length parameterized plane curve can be expressed as the rate of change in the tangent Equivalently: Here the angle brackets denote the usual dot product, i.e., . Review: Curvature and Torsion of a Space Curve •For a plane curve, curvature captured the notion of “bending” •For a space curve we also have torsion, which captures “twisting” Intuition: torsion is “out of plane bending” increasing torsion Review: Fundamental Theorem of Space Curves •The fundamental theorem of space curves tells that given the curvature κ and torsion τ of an arc-length parameterized space curve, we can recover the curve (up to rigid motion) •Formally: integrate the Frenet-Serret equations; intuitively: start drawing a curve, bend & twist at prescribed rate. -
Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature
Annals of Mathematics Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature Author(s): William Meeks III, Leon Simon and Shing-Tung Yau Source: Annals of Mathematics, Second Series, Vol. 116, No. 3 (Nov., 1982), pp. 621-659 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/2007026 Accessed: 06-02-2018 14:54 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics This content downloaded from 129.93.180.106 on Tue, 06 Feb 2018 14:54:09 UTC All use subject to http://about.jstor.org/terms Annals of Mathematics, 116 (1982), 621-659 Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature By WILLIAM MEEKS III, LEON SIMON and SHING-TUNG YAU Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some "canonical" embedded surface. From the point of view of geometry, a natural "canonical" surface will be the extremal surface of some functional defined on the space of embedded surfaces. -
Scalable Synthesis of Gyroid-Inspired Freestanding Three-Dimensional Graphene Architectures† Cite This: DOI: 10.1039/C9na00358d Adrian E
Nanoscale Advances View Article Online COMMUNICATION View Journal Scalable synthesis of gyroid-inspired freestanding three-dimensional graphene architectures† Cite this: DOI: 10.1039/c9na00358d Adrian E. Garcia,a Chen Santillan Wang, a Robert N. Sanderson,b Kyle M. McDevitt,a c ac a d Received 5th June 2019 Yunfei Zhang, Lorenzo Valdevit, Daniel R. Mumm, Ali Mohraz a Accepted 16th September 2019 and Regina Ragan * DOI: 10.1039/c9na00358d rsc.li/nanoscale-advances Three-dimensional porous architectures of graphene are desirable for Introduction energy storage, catalysis, and sensing applications. Yet it has proven challenging to devise scalable methods capable of producing co- Porous 3D architectures composed of graphene lms can continuous architectures and well-defined, uniform pore and liga- improve performance of carbon-based scaffolds in applications Creative Commons Attribution-NonCommercial 3.0 Unported Licence. ment sizes at length scales relevant to applications. This is further such as electrochemical energy storage,1–4 catalysis,5 sensing,6 complicated by processing temperatures necessary for high quality and tissue engineering.7,8 Graphene's remarkably high elec- graphene. Here, bicontinuous interfacially jammed emulsion gels trical9 and thermal conductivities,10 mechanical strength,11–13 (bijels) are formed and processed into sacrificial porous Ni scaffolds for large specic surface area,14 and chemical stability15 have led to chemical vapor deposition to produce freestanding three-dimensional numerous explorations of this multifunctional material due to turbostratic graphene (bi-3DG) monoliths with high specificsurface its promise to impact multiple elds. While these specic area. Scanning electron microscopy (SEM) images show that the bi- applications typically require porous 3D architectures, gra- 3DG monoliths inherit the unique microstructural characteristics of phene growth has been most heavily investigated on 2D their bijel parents. -
Lord Kelvin's Error? an Investigation Into the Isotropic Helicoid
WesleyanUniversity PhysicsDepartment Lord Kelvin’s Error? An Investigation into the Isotropic Helicoid by Darci Collins Class of 2018 An honors thesis submitted to the faculty of Wesleyan University in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Departmental Honors in Physics Middletown,Connecticut April,2018 Abstract In a publication in 1871, Lord Kelvin, a notable 19th century scientist, hypothesized the existence of an isotropic helicoid. He predicted that such a particle would be isotropic in drag and rotation translation coupling, and also have a handedness that causes it to rotate. Since this work was published, theorists have made predictions about the motion of isotropic helicoids in complex flows. Until now, no one has built such a particle or quantified its rotation translation coupling to confirm whether the particle has the properties that Lord Kelvin predicted. In this thesis, we show experimental, theoretical, and computational evidence that all conclude that Lord Kelvin’s geometry of an isotropic helicoid does not couple rotation and translation. Even in both the high and low Reynolds number regimes, Lord Kelvin’s model did not rotate through fluid. While it is possible there may be a chiral particle that is isotropic in drag and rotation translation coupling, this thesis presents compelling evidence that the geometry Lord Kelvin proposed is not one. Our evidence leads us to hypothesize that an isotropic helicoid does not exist. Contents 1 Introduction 2 1.1 What is an Isotropic Helicoid? . 2 1.1.1 Lord Kelvin’s Motivation . 3 1.1.2 Mentions of an Isotropic Helicoid in Literature . -
Optical Properties of Gyroid Structured Materials: from Photonic Crystals to Metamaterials
REVIEW ARTICLE Optical properties of gyroid structured materials: from photonic crystals to metamaterials James A. Dolan1;2;3, Bodo D. Wilts2;4, Silvia Vignolini5, Jeremy J. Baumberg3, Ullrich Steiner2;4 and Timothy D. Wilkinson1 1 Department of Engineering, University of Cambridge, JJ Thomson Avenue, CB3 0FA, Cambridge, United Kingdom 2 Cavendish Laboratory, Department of Physics, University of Cambridge, JJ Thomson Avenue, CB3 0HE, Cambridge, United Kingdom 3 NanoPhotonics Centre, Cavendish Laboratory, Department of Physics, University of Cambridge, JJ Thomson Avenue, CB3 0HE, Cambridge, United Kingdom 4 Adolphe Merkle Institute, University of Fribourg, Chemin des Verdiers 4, CH-1700 Fribourg, Switzerland 5 Department of Chemistry, University of Cambridge, Lensfield Road, CB2 1EW, Cambridge, United Kingdom E-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] Abstract. The gyroid is a continuous and triply periodic cubic morphology which possesses a constant mean curvature surface across a range of volumetric fill fractions. Found in a variety of natural and synthetic systems which form through self-assembly, from butterfly wing scales to block copolymers, the gyroid also exhibits an inherent chirality not observed in any other similar morphologies. These unique geometrical properties impart to gyroid structured materials a host of interesting optical properties. Depending on the length scale on which the constituent materials are organised, these properties arise from starkly different physical mechanisms (such as a complete photonic band gap for photonic crystals and a greatly depressed plasma frequency for optical metamaterials). This article reviews the theoretical predictions and experimental observations of the optical properties of two fundamental classes of gyroid structured materials: photonic crystals (wavelength scale) and metamaterials (sub- wavelength scale). -
Curvatures of Smooth and Discrete Surfaces
Curvatures of Smooth and Discrete Surfaces John M. Sullivan The curvatures of a smooth curve or surface are local measures of its shape. Here we consider analogous quantities for discrete curves and surfaces, meaning polygonal curves and triangulated polyhedral surfaces. We find that the most useful analogs are those which preserve integral relations for curvature, like the Gauß/Bonnet theorem. For simplicity, we usually restrict our attention to curves and surfaces in euclidean three space E3, although many of the results would easily generalize to other ambient manifolds of arbitrary dimension. Most of the material here is not new; some is even quite old. Although some refer- ences are given, no attempt has been made to give a comprehensive bibliography or an accurate picture of the history of the ideas. 1. Smooth curves, Framings and Integral Curvature Relations A companion article [Sul06] investigated the class of FTC (finite total curvature) curves, which includes both smooth and polygonal curves, as a way of unifying the treatment of curvature. Here we briefly review the theory of smooth curves from the point of view we will later adopt for surfaces. The curvatures of a smooth curve γ (which we usually assume is parametrized by its arclength s) are the local properties of its shape invariant under Euclidean motions. The only first-order information is the tangent line; since all lines in space are equivalent, there are no first-order invariants. Second-order information is given by the osculating circle, and the invariant is its curvature κ = 1/r. For a plane curve given as a graph y = f(x) let us contrast the notions of curvature and second derivative. -
Differential Geometry: Curvature and Holonomy Austin Christian
University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics. -
Surfaces Minimales : Theorie´ Variationnelle Et Applications
SURFACES MINIMALES : THEORIE´ VARIATIONNELLE ET APPLICATIONS FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) Contents 1. Introduction 1 2. First variation formula 1 3. Examples 4 4. Maximum principle 5 5. Calibration: area-minimizing surfaces 6 6. Second Variation Formula 8 7. Monotonicity Formula 12 8. Bernstein Theorem 16 9. The Stability Condition 18 10. Simons' Equation 29 11. Schoen-Simon-Yau Theorem 33 12. Pointwise curvature estimates 38 13. Plateau problem 43 14. Harmonic maps 51 15. Positive Mass Theorem 52 16. Harmonic Maps - Part 2 59 17. Ricci Flow and Poincar´eConjecture 60 18. The proof of Willmore Conjecture 65 1. Introduction 2. First variation formula Let (M n; g) be a Riemannian manifold and Σk ⊂ M be a submanifold. We use the notation jΣj for the volume of Σ. Consider (x1; : : : ; xk) local coordinates on Σ and let @ @ gij(x) = g ; ; for 1 ≤ i; j ≤ k; @xi @xj 1 2 FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) be the components of gjΣ. The volume element of Σ is defined to be the p differential k-form dΣ = det gij(x). The volume of Σ is given by Z Vol(Σ) = dΣ: Σ Consider the variation of Σ given by a smooth map F :Σ × (−"; ") ! M. Use Ft(x) = F (x; t) and Σt = Ft(Σ). In this section we are interested in the first derivative of Vol(Σt). Definition 2.1 (Divergence). Let X be a arbitrary vector field on Σk ⊂ M. We define its divergence as k X (1) divΣX(p) = hrei X; eii; i=1 where fe1; : : : ; ekg ⊂ TpΣ is an orthonormal basis and r is the Levi-Civita connection with respect to the Riemannian metric g. -
Quintic Parametric Polynomial Minimal Surfaces and Their Properties
Differential Geometry and its Applications 28 (2010) 697–704 Contents lists available at ScienceDirect Differential Geometry and its Applications www.elsevier.com/locate/difgeo Quintic parametric polynomial minimal surfaces and their properties ∗ Gang Xu a,c, ,Guo-zhaoWangb a Hangzhou Dianzi University, Hangzhou 310018, PR China b Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China c Galaad, INRIA Sophia-Antipolis, 2004 Route des Lucioles, 06902 Sophia-Antipolis Cedex, France article info abstract Article history: In this paper, quintic parametric polynomial minimal surface and their properties are Received 28 November 2009 discussed. We first propose the sufficient condition of quintic harmonic polynomial Received in revised form 14 April 2010 parametric surface being a minimal surface. Then several new models of minimal surfaces Available online 31 July 2010 with shape parameters are derived from this condition. We also study the properties Communicated by Z. Shen of new minimal surfaces, such as symmetry, self-intersection on symmetric planes and MSC: containing straight lines. Two one-parameter families of isometric minimal surfaces are 53A10 also constructed by specifying some proper shape parameters. 49Q05 © 2010 Elsevier B.V. All rights reserved. 49Q10 Keywords: Minimal surface Harmonic surfaces Isothermal parametric surface Isometric minimal surfaces 1. Introduction Since Lagrange derived the minimal surface equation in R3 in 1762, minimal surfaces have a long history of over 200 years. A minimal surface is a surface with vanishing mean curvature. As the mean curvature is the variation of area functional, minimal surfaces include the surfaces minimizing the area with a fixed boundary. Because of their attractive properties, minimal surfaces have been extensively employed in many areas such as architecture, material science, aviation, ship manufacture, biology and so on. -
Generating Carbon Schwarzites Via Zeolite-Templating
Generating carbon schwarzites via zeolite-templating Efrem Brauna, Yongjin Leeb,c, Seyed Mohamad Moosavib, Senja Barthelb, Rocio Mercadod, Igor A. Baburine, Davide M. Proserpiof,g, and Berend Smita,b,1 aDepartment of Chemical and Biomolecular Engineering, University of California, Berkeley, CA 94720; bInstitut des Sciences et Ingenierie´ Chimiques (ISIC), Valais, Ecole´ Polytechnique Fed´ erale´ de Lausanne (EPFL), CH-1951 Sion, Switzerland; cSchool of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China; dDepartment of Chemistry, University of California, Berkeley, CA 94720; eTheoretische Chemie, Technische Universitat¨ Dresden, 01062 Dresden, Germany; fDipartimento di Chimica, Universita` degli Studi di Milano, 20133 Milan, Italy; and gSamara Center for Theoretical Materials Science (SCTMS), Samara State Technical University, Samara 443100, Russia Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved July 9, 2018 (received for review March 25, 2018) Zeolite-templated carbons (ZTCs) comprise a relatively recent ZTC can be seen as a schwarzite (24–28). Indeed, the experi- material class synthesized via the chemical vapor deposition of mental properties of ZTCs are exactly those which have been a carbon-containing precursor on a zeolite template, followed predicted for schwarzites, and hence ZTCs are considered as by the removal of the template. We have developed a theoret- promising materials for the same applications (21, 22). As we ical framework to generate a ZTC model from any given zeolite will show, the similarity between ZTCs and schwarzites is strik- structure, which we show can successfully predict the structure ing, and we explore this similarity to establish that the theory of known ZTCs. We use our method to generate a library of of schwarzites/TPMSs is a useful concept to understand ZTCs. -
Minimal Surfaces Saint Michael’S College
MINIMAL SURFACES SAINT MICHAEL’S COLLEGE Maria Leuci Eric Parziale Mike Thompson Overview What is a minimal surface? Surface Area & Curvature History Mathematics Examples & Applications http://www.bugman123.com/MinimalSurfaces/Chen-Gackstatter-large.jpg What is a Minimal Surface? A surface with mean curvature of zero at all points Bounded VS Infinite A plane is the most trivial minimal http://commons.wikimedia.org/wiki/File:Costa's_Minimal_Surface.png surface Minimal Surface Area Cube side length 2 Volume enclosed= 8 Surface Area = 24 Sphere Volume enclosed = 8 r=1.24 Surface Area = 19.32 http://1.bp.blogspot.com/- fHajrxa3gE4/TdFRVtNB5XI/AAAAAAAAAAo/AAdIrxWhG7Y/s160 0/sphere+copy.jpg Curvature ~ rate of change More Curvature dT κ = ds Less Curvature History Joseph-Louis Lagrange first brought forward the idea in 1768 Monge (1776) discovered mean curvature must equal zero Leonhard Euler in 1774 and Jean Baptiste Meusiner in 1776 used Lagrange’s equation to find the first non- trivial minimal surface, the catenoid Jean Baptiste Meusiner in 1776 discovered the helicoid Later surfaces were discovered by mathematicians in the mid nineteenth century Soap Bubbles and Minimal Surfaces Principle of Least Energy A minimal surface is formed between the two boundaries. The sphere is not a minimal surface. http://arxiv.org/pdf/0711.3256.pdf Principal Curvatures Measures the amount that surfaces bend at a certain point Principal curvatures give the direction of the plane with the maximal and minimal curvatures http://upload.wikimedia.org/wi