|
Gyroid
A gyroid is an infinitely connected 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 P and 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 embedded, and in 1997 Große-Brauckmann provided CMC ( constant mean curvature) variants of the gyroid and made further numerical investigations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroid Unit Cell
A gyroid is an infinitely connected 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 P and 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 embedded, and in 1997 Große-Brauckmann provided CMC (constant mean curvature) variants of the gyroid and made further numerical investigations abo ... [...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] |
Photonic Crystals
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic lattices (crystal structure) of semiconductors affect their conductivity of electrons. Photonic crystals occur in nature in the form of structural coloration and animal reflectors, and, as artificially produced, promise to be useful in a range of applications. Photonic crystals can be fabricated for one, two, or three dimensions. One-dimensional photonic crystals can be made of thin film layers deposited on each other. Two-dimensional ones can be made by photolithography, or by drilling holes in a suitable substrate. Fabrication methods for three-dimensional ones include drilling under different angles, stacking multiple 2-D layers on top of each other, direct laser writing, or, for example, instigating self-assembly of spheres in a ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laves Graph
In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance \sqrt2. It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices. named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932. It has also been called the ''K''4 crystal, (10,3)-a network, diamond twin, triamond, and the srs net. The regions of space nearest each vertex of the graph are 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 cal ...: :\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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 divi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Copolymer
In polymer chemistry, a copolymer is a polymer derived from more than one species of monomer. The polymerization of monomers into copolymers is called copolymerization. Copolymers obtained from the copolymerization of two monomer species are sometimes called ''bipolymers''. Those obtained from three and four monomers are called ''terpolymers'' and ''quaterpolymers'', respectively. Copolymers can be characterized by a variety of techniques such as NMR spectroscopy and size-exclusion chromatography to determine the molecular size, weight, properties, and composition of the material. Commercial copolymers include acrylonitrile butadiene styrene (ABS), styrene/butadiene co-polymer (SBR), nitrile rubber, styrene-acrylonitrile, styrene-isoprene-styrene (SIS) and ethylene-vinyl acetate, all of which are formed by chain-growth polymerization. Another production mechanism is step-growth polymerization, which is used to produce the nylon-12/6/66 copolymer of nylon 12, nylon 6 and n ... [...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 nanostr ... [...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 The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant-mean-curvature Surface
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2003:E1/ref> This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. History In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid. In 1853 J. H. Jellet showed that if S is a compact star-shaped surface in \R^3 with constant mean curvature, then it is the standard sphere. Subsequently, A. D. Alexandrov proved that a compact embedded surface in \R^3 with constant mean curvature H \neq 0 must ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
