Or: Selected Classification Problems in Minimal Surface Theory
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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. -
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 . -
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. -
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). -
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 -