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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] |
Lidinoid Surface
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 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), .... It belongs to space group 230(Ia3d). The Lidinoid can be approximated as a level set: :\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 minima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...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 divid ... [...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] |
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] |
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] |
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] |
Space Group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups. In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the ''International Tables for Crystallography'' . History Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 called a level curve, also known as ''contour line'' or ''isoline''; so a level curve is the set of all real-valued solutions of an equation in two variables and . When , a level set is called a level surface (or ''isosurface''); so a level surface is the set of all real-valued roots of an equation in three variables , and . For higher values of , the level set is a level hypersurface, the set of all real-valued roots of an equation in variables. A level set is a special case of a fiber. Alternative names Level sets show up in many applications, often under different names. For example, an implicit curve is a level curve, which is considered independently of its neighbor curves, emphasizing that such a curve is defined by an implicit e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surfaces
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] |
