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Plateau's Rules
Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws. Laws for soap films Plateau's laws describe the shape and configuration of soap films as follows: # Soap films are made of entire (unbroken) smooth surfaces. # The mean curvature of a portion of a soap film is everywhere constant on any point on the same piece of soap film. # Soap films always meet in threes along an edge called a Plateau border, and they do so at an angle of arccos(−) = 120°. # These Plateau borders meet in fours at a vertex, at the tetrahedral angle of arccos(−) ≈ 109.47°. Configurations other than those of Plateau's laws are unstable, and the film will quickly tend to rearrange itself to conform to these laws. That these laws hold for minimal surfaces was proved mathematically by Jean Taylor using geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soap Bubbles (41493399275)
Soap is a salt of a fatty acid used in a variety of cleansing and lubricating products. In a domestic setting, soaps are surfactants usually used for washing, bathing, and other types of housekeeping. In industrial settings, soaps are used as thickeners, components of some lubricants, and precursors to catalysts. When used for cleaning, soap solubilizes particles and grime, which can then be separated from the article being cleaned. In hand washing, as a surfactant, when lathered with a little water, soap kills microorganisms by disorganizing their membrane lipid bilayer and denaturing their proteins. It also emulsifies oils, enabling them to be carried away by running water. Soap is created by mixing fats and oils with a base. A similar process is used for making detergent which is also created by combining chemical compounds in a mixer. Humans have used soap for millennia. Evidence exists for the production of soap-like materials in ancient Babylon around 2800 BC. Type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soap Film
Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Plateau borders. Soap films can be used as model systems for minimal surfaces, which are widely used in mathematics. Stability Daily experience shows that soap bubble formation is not feasible with water or with any pure liquid. Actually, the presence of soap, which is composed at a molecular scale of surfactants, is necessary to stabilize the film. Most of the time, surfactants are amphiphilic, which means they are molecules with both a hydrophobic and a hydrophilic part. Thus, they are arranged preferentially at the air/water interface (see figure 1). Surfactants stabilize films because they create a repulsion between both surfaces of the film, preventing it from thinning and consequentially bursting. This can be shown quantitatively thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belgium
Belgium, ; french: Belgique ; german: Belgien officially the Kingdom of Belgium, is a country in Northwestern Europe. The country is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to the southwest, and the North Sea to the northwest. It covers an area of and has a population of more than 11.5 million, making it the 22nd most densely populated country in the world and the 6th most densely populated country in Europe, with a density of . Belgium is part of an area known as the Low Countries, historically a somewhat larger region than the Benelux group of states, as it also included parts of northern France. The capital and largest city is Brussels; other major cities are Antwerp, Ghent, Charleroi, Liège, Bruges, Namur, and Leuven. Belgium is a sovereign state and a federal constitutional monarchy with a parliamentary system. Its institutional organization is complex and is structured on both regional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Plateau
Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist and mathematician. He was one of the first people to demonstrate the illusion of a moving image. To do this, he used counterrotating disks with repeating drawn images in small increments of motion on one and regularly spaced slits in the other. He called this device of 1832 the phenakistiscope. Biography Plateau was born in Brussels. His father, Antoine Plateau ( fr) born in Tournai, was a talented flower painter. At the age of six, the younger Plateau already could read, making him a child prodigy in those times. While attending primary school, he was particularly impressed by a lesson of physics; enchanted by the experiments he observed, he vowed to discover their secrets someday. Plateau spent his school holidays in Marche-les-Dames, with his uncle and his family; his cousin and playfellow was Auguste Payen, who later became an architect and the principal designer of the Belgian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patterns In Nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time. In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature
In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation. Definition Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that curvatur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Angle
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Taylor
Jean Ellen Taylor (born 1944) is an American mathematician who is a professor emerita at Rutgers University and visiting faculty at Courant Institute of Mathematical Sciences of New York University. Biography Taylor was born in Northern California. She did her undergraduate studies at Mount Holyoke College, graduating summa cum laude with an A.B. in 1966. She began her graduate studies in chemistry at the University of California, Berkeley, but after receiving an M.Sc. she switched to mathematics under the mentorship of S. S. Chern and then transferred to the University of Warwick and received a second M.Sc. in mathematics there. She completed a doctorate in 1973 from Princeton University under the supervision of Frederick J. Almgren, Jr. Taylor joined the Rutgers faculty in 1973, and retired in 2002. She was president of the Association for Women in Mathematics from 1999 to 2001. She has been married three times, to mathematicians John Guckenheimer and Fred Almgren, and to fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Measure Theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth. History Geometric measure theory was born out of the desire to solve Plateau's problem (named after Joseph Plateau) which asks if for every smooth closed curve in \mathbb^3 there exists a surface of least area among all surfaces whose boundary equals the given curve. Such surfaces mimic soap films. The problem had remained open since it was posed in 1760 by Lagrange. It was solved independently in the 1930s by Jesse Douglas and Tibor Radó under certain topological restrictions. In 1960 Herbert Federer and Wendell Fleming used the theory of currents with which they were able to solve the orientable Plateau's problem analytically without topological restrictions, thus sparking geometric measure the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it is the oldest continuously published magazine in the United States. ''Scientific American'' is owned by Springer Nature, which in turn is a subsidiary of Holtzbrinck Publishing Group. History ''Scientific American'' was founded by inventor and publisher Rufus Porter (painter), Rufus Porter in 1845 as a four-page weekly newspaper. The first issue of the large format newspaper was released August 28, 1845. Throughout its early years, much emphasis was placed on reports of what was going on at the United States Patent and Trademark Office, U.S. Patent Office. It also reported on a broad range of inventions including perpetual motion machines, an 1860 device for buoying vessels by Abraham Lincoln, and the universal joint which now can be found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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