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Structural Rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorics, combinatorial theory for predicting the flexibility of ensembles formed by rigid body, rigid bodies connected by flexible Linkage (mechanical), linkages or hinges. Definitions Stiffness, Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges. There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a ...
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Structural Rigidity Basic Examples
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as organism, biological organisms, minerals and chemical substance, chemicals. Abstract structures include data structures in computer science and musical form. Types of structure include a hierarchy (a cascade of one-to-many relationships), a Complex network, network featuring many-to-many Link (geometry), links, or a lattice (order), lattice featuring connections between components that are neighbors in space. Load-bearing Buildings, aircraft, skeletons, Ant colony, anthills, beaver dams, bridges and salt domes are all examples of Structural load, load-bearing structures. The results of construction are divided into buildings and nonbuilding structure, non-building structures, and make up the infrastructure of a human society. Built str ...
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Cauchy's Theorem (geometry)
Cauchy's theorem is a theorem in geometry, named after Augustin-Louis Cauchy, Augustin Cauchy. It states that convex polytopes in three dimensions with congruence (geometry), congruent corresponding faces must be congruent to each other. That is, any Net (polyhedron), polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape. This is a fundamental result in rigidity theory (structural), rigidity theory: one consequence of the theorem is that, if one makes a physical model of a convex polyhedron by connecting together rigid plates for each of the polyhedron faces with flexible hinges along the polyhedron ...
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Counting On Frameworks
''Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures'' is an undergraduate-level book on the mathematics of structural rigidity. It was written by Jack E. Graver and published in 2001 by the Mathematical Association of America as volume 25 of the Dolciani Mathematical Expositions book series. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion by undergraduate mathematics libraries. Topics The problems considered by ''Counting on Frameworks'' primarily concern systems of rigid rods, connected to each other by flexible joints at their ends; the question is whether these connections fix such a framework into a single position, or whether it can flex continuously through multiple positions. Variations of this problem include the simplest way to add rods to a framework to make it rigid, or the resiliance of a framework against the failure of one of its rods. To study this question, Graver has organized ''Coun ...
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Chebychev–Grübler–Kutzbach Criterion
The Pafnuty Chebyshev, Chebychev–Grübler–Kutzbach criterion determines the number of degrees of freedom of a kinematic chain, that is, a coupling of rigid bodies by means of mechanical constraints. These devices are also called linkage (mechanical), linkages. The Kutzbach criterion is also called the ''mobility formula'', because it computes the number of parameters that define the configuration of a linkage from the number of links and joints and the degree of freedom at each joint. Interesting and useful linkages have been designed that violate the mobility formula by using special geometric features and dimensions to provide more mobility than predicted by this formula. These devices are called overconstrained mechanisms. Mobility formula The mobility formula counts the number of parameters that define the positions of a set of rigid bodies and then reduces this number by the constraints that are imposed by joints connecting these bodies.J. J. Uicker, G. R. Pennock, a ...
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James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ... and scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. It involves ... responsible for the classical theory Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift A paradi ... of electromagnetic radiation In physics Physics is the natural science that s ...
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Bipartite Graph
In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ... field of graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ..., a bipartite graph (or bigraph) is a graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ... whose vertices can be divided into two disjoint and independent sets U and V such that every edge Edge or EDGE may refer to: Technology Computi ...
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Cross Bracing
In construction, cross bracing is a system utilized to reinforce building structures in which diagonal supports intersect. Cross bracing is usually seen with two diagonal supports placed in an X-shaped manner. Under lateral force (such as wind or Earthquake, seismic activity) one brace will be under Tension (physics), tension while the other is being Compression (physics), compressed. In steel construction, steel cables may be used due to their great resistance to tension (although they cannot take any load in compression (physical), compression). The common uses for cross bracing include bridge (side) supports, along with structural foundations. This method of construction maximizes the weight of the load a structure is able to support. It is a usual application when constructing earthquake-safe buildings. Cross bracing can be applied to any rectangular frame structure, such as chairs and bookshelves. Its rigidity for two-dimensional grid structures can be analyzed mathematically ...
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Square Grid
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 Square (geometry), squares around every Vertex (geometry), vertex. John Horton Conway, Conway called it a quadrille. The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of List of regular polytopes#Euclidean tilings, three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. Related polyhedra and tilings Th ...
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Grid Bracing
In the mathematics of structural rigidity, grid bracing is a problem of adding cross bracing to a square grid to make it into a rigid structure. It can be solved optimally by translating it into a problem in graph theory on the Connectivity (graph theory), connectivity of bipartite graphs. Problem statement The problem considers a framework in the form of a square grid, with r rows and c columns of squares. The grid has r(c+1)+(r+1)c edges, each of which has unit length and is considered to be a rigid rod, free to move continuously within the Euclidean plane but unable to change its length. These rods are attached to each other by flexible joints at the (r+1)(c+1) vertices of the grid. A valid continuous motion of this framework is a way of continuously varying the placement of its edges and joints into the plane in such a way that they keep the same lengths and the same attachments, but without requiring them to form squares. Instead, each square of the grid may be deformed to form ...
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Volume
Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact of existence. The term has a history of use reaching ba ... of three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ... enclosed by a closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree .... For example, the space that a substance (solid Solid is one of the four fundamental states of matter (the others be ...
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Bellows Conjecture
In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy's theorem (geometry), Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex set, convex (this is also true in higher dimensions). The first examples of flexible polyhedra, now called Bricard octahedron, Bricard octahedra, were discovered by . They are self-intersecting surfaces isometry, isometric to an octahedron. The first example of a flexible non-self-intersecting surface in \mathbb^3, the Connelly sphere, was discovered by . Steffen's polyhedron is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra. Bellows conjecture In the late 1970s Connelly and D. Sullivan formulated the bellows conjecture stating that the volume of a flexible polyhedron is invariant (mathematics), invariant under flexing. This conjecture wa ...
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Robert Connelly
Robert Connelly (born July 15, 1942) is a mathematician specializing in discrete geometry and rigidity theory (structural), rigidity theory. Connelly received his Ph.D. from University of Michigan in 1969. He is currently a professor at Cornell University. Connelly is best known for discovering embedded flexible polyhedra. One such polyhedron is in the National Museum of American History. His recent interests include Tensegrity, tensegrities and the carpenter's ruler problem, carpenter's rule problem. In 2012 he became a fellow of the American Mathematical Society. Asteroid 4816 Connelly, discovered by Edward Bowell at Lowell Observatory 1981, was named after Robert Connelly. The official was published by the Minor Planet Center on 18 February 1992 (). Author Connelly has authored or co-authored several articles on mathematics, including ''Conjectures and open questions in rigidity''; ''A flexible sphere''; and ''A counterexample to the rigidity conjecture for polyhedra''. ...
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