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Triply Periodic Minimal Surface
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise. TPMS are of relevance in natural science. TPMS have been observed as biological membranes, as block copolymers, equipotential surfaces in crystals etc. They have also been of interest in architecture, design and art. Properties Nearly all studied TPMS are free of self-intersections (i.e. embedded in ℝ3): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant). All connected TPMS have genus ≥ 3, and in every lattice there exist orientable embedded TPMS of every genus ≥3. Embedded TPMS are orientable and divid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz H Surface
Schwarz may refer to: * Schwarz, Germany, a municipality in Mecklenburg-Vorpommern, Germany * Schwarz (surname), a surname (and list of people with the surname) * Schwarz (musician), American DJ and producer * ''Schwarz'' (Böhse Onkelz album), released simultaneously with ''Weiß'', 1993 * ''Schwarz'' (Conrad Schnitzler album), a reissue of the 1971 Kluster album ''Eruption'' * Schwarz (cards), in some card games, a Schneider (low point score) in which no tricks are taken * Schwarz Gruppe, a multinational retail group * Schwarz Pharma, a German drug company See also * * * Schwartz (other) Schwartz may refer to: * Schwartz (surname), a surname (and list of people with the name) * Schwartz (brand), a spice brand *Schwartz's, a delicatessen in Montreal, Quebec, Canada * Schwartz Publishing, an Australian publishing house *"Danny Schwar ... * Schwarzhorn (other) * Swartz (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Minimal Surface
In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz. In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named by Alan Schoen in his seminal report that described the gyroid and other triply periodic minimal surfaces. The surfaces were generated using symmetry arguments: given a solution to Plateau's problem for a polygon, reflections of the surface across the boundary lines also produce valid minimal surfaces that can be continuously joined to the original solution. If a minimal surface meets a plane at right angles, then the mirror image in the plane can also be joined to the surface. Hence given a suitable initial polygon inscribed in a unit cell periodic surfaces can be constructed. The Schwarz surfaces have topological genus 3, the minimal genus of triply periodic minimal surfaces. They have been considered as models for periodic nanostructu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasicrystal
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders—for instance, five-fold. Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of natural quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the field of crystallography. In crystallography the quasicrystals were predicted in 1981 by a five-fold symmetry study of Alan Lindsay Mackay,—that also brought in 1982, with the crystallographic Fourier transform of a Penrose tiling,Alan L. Mackay, "Crystallography ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiperiodic
Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpredictability that does not lend itself to precise measurement. It is different from the mathematical concept of an almost periodic function, which has increasing regularity over multiple periods. The mathematical definition of quasiperiodic function is a completely different concept; the two should not be confused. Climatology Climate oscillations that appear to follow a regular pattern but which do not have a fixed period are called ''quasiperiodic''. Within a dynamical system such as the ocean-atmosphere oscillations may occur regularly, when they are forced by a regular external forcing: for example, the familiar winter-summer cycle is forced by variations in sunlight from the (very close to perfectly) periodic motion of the earth around ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joan Hutchinson
Joan Prince Hutchinson (born 1945) is an American mathematician and Professor Emerita of Mathematics from Macalester College. Education Joan Hutchinson was born in Philadelphia, Pennsylvania; her father was a demographer and university professor, and her mother a mathematics teacher at the Baldwin School, which Joan also attended. She studied at Smith College in Northampton, Massachusetts, graduating in 1967 summa cum laude with an honors paper directed by Prof. Alice Dickinson. After graduation she worked as a computer programmer at the Woods Hole Oceanographic Institute and at the Harvard University Computing Center then studied mathematics (and English change ringing on tower bells) at the University of Warwick in Coventry England. Returning to the United States, Hutchinson did graduate work at the University of Pennsylvania earning a Ph.D. in mathematics in 1973 under the supervision of Herbert S. Wilf. Career She was a John Wesley Young research instructor at Dartmouth Col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerhard Huisken
Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity. Education and career After finishing high school in 1977, Huisken took up studies in mathematics at Heidelberg University. In 1982, one year after his diploma graduation, he completed his PhD at the same university under the direction of Claus Gerhardt. The topic of his dissertation were non-linear partial differential equations (''Reguläre Kapillarflächen in negativen Gravitationsfeldern''). From 1983 to 1984, Huisken was a researcher at the Centre for Mathematical Analysis at the Australian National University (ANU) in C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lidinoid
In differential geometry, the lidinoid is a triply periodic minimal surface. The name comes from its Swedish discoverer Sven Lidin (who called it the HG surface). It has many similarities to the gyroid, and just as the gyroid is the unique embedded member of the associate family of the Schwarz P surface the lidinoid is the unique embedded member of the associate family of a Schwarz H surface. It belongs to space group 230(Ia3d). The Lidinoid can be approximated as a level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...: :\begin (1/2) \sin(2x) \cos(y)\sin(z)\\ + &\sin(2y)\cos(z) \sin(x)\\ + &\sin(2z)\cos(x) \sin(y)\ -& (1/2) cos(2x)\cos(2y)\\ + &\cos(2y)\cos(2z)\\ + &\cos(2z)\cos(2x) + 0.15 = 0 \end References External images The lidinoid at the minimal su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroid
A gyroid is an infinitely connected Triply periodic minimal surface, triply periodic minimal surface discovered by Alan Schoen in 1970. History and properties The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz minimal surface#Schwarz P .28.22Primitive.22.29, Schwarz P and Schwarz minimal surface#Schwarz D .28.22Diamond.22.29, D surfaces. Its angle of association with respect to the D surface is approximately 38.01°. The gyroid is similar to the lidinoid. The gyroid was discovered in 1970 by NASA scientist Alan Schoen. He calculated the angle of association and gave a convincing demonstration of pictures of intricate plastic models, but did not provide a proof of embeddedness. Schoen noted that the gyroid contains neither straight lines nor planar symmetries. Karcher gave a different, more contemporary treatment of the surface in 1989 using conjugate surface construction. In 1996 Große-Brauckmann and Wohlgemuth proved that it is embed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Differential Geometry
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, geometry processing and topological combinatorics. See also *Discrete Laplace operator *Discrete exterior calculus *Discrete Morse theory *Topological combinatorics *Spectral shape analysis * Abstract differential geometry *Analysis on fractals *Discrete calculus Discrete calculus or the calculus of discrete functions, is the mathematical study of ''incremental'' change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word ''ca ... ReferencesDiscrete differential geometry Forum* * * Alexander I. Bobenko, Yuri B. Suris (2008), "Discrete Differential Geometry", American Mathematical Society, Differential geometry Simplicial sets {{differential-geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass–Enneper Parameterization
In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863. Let f and g be functions on either the entire complex plane or the unit disk, where g is meromorphic and f is analytic, such that wherever g has a pole of order m, f has a zero of order 2m (or equivalently, such that the product f g^2 is holomorphic), and let c_1,c_2,c_3 be constants. Then the surface with coordinates (x_1, x_2, x_3) is minimal, where the x_k are defined using the real part of a complex integral, as follows: \begin x_k(\zeta) &= \Re \left\ + c_k , \qquad k=1,2,3 \\ \varphi_1 &= f(1-g^2)/2 \\ \varphi_2 &= \mathbf f(1+g^2)/2 \\ \varphi_3 &= fg \end The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type. For example, Enneper's surface has , . Parametric surface of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associate Family
In differential geometry, the associate family (or Bonnet family) of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has the representation :x_k(\zeta) = \Re \left\ + c_k , \qquad k=1,2,3 the family is described by :x_k(\zeta,\theta) = \Re \left\ + c_k , \qquad \theta \in ,2\pi where \Re indicates the real part of a complex number. For ''θ'' = ''π''/2 the surface is called the conjugate of the ''θ'' = 0 surface. The transformation can be viewed as locally rotating the principal curvature directions. The surface normals of a point with a fixed ''ζ'' remains unchanged as ''θ'' changes; the point itself moves along an ellipse. Some examples of associate surface families are: the catenoid and helicoid family, the Schwarz P, Schwarz D and gyroid family, and the Scherk's first and second surface family. The Enneper surface In differential geometry and algebraic geometry, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Schoen
Alan Hugh Schoen (born December 11, 1924) is an American physicist and computer scientist best known for his discovery of the gyroid, an infinitely connected triply periodic minimal surface. Professional career Alan Schoen received his B.S. degree in physics from Yale University in 1945 and his Ph.D. in physics from University of Illinois at Urbana-Champaign in 1958.Alan Schoen. 2017. Personal email communication with J. Kocik His doctoral dissertation was entitled “Self-Diffusion in Alpha Solid Solutions of Silver-Cadmium and Silver-Indium.” After completing graduate work he was employed (between 1957 and 1967) as a research physicist by aerospace companies in California, and also worked as a free-lance solid-state physics consultant. In 1967, he took the position of senior scientist at NASA's Electronics Research Center (ERC) in Cambridge, Massachusetts,Alan H. Schoen. Triply-periodic minimal surfaces. http://schoengeometry.com/e-tpms.html where he did geometry research and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
