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Flexible Polyhedra
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 rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this is also true in higher dimensions). The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by . They are self-intersecting surfaces 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 under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by using elimination theory, and then pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
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 sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David A
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the third king of the United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges a notably close friendship with Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistines, a 30-year-old David is anointed king over all of Israel and Judah. Following his rise to power, David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background. Paid circulation in 2008 was 9,500 and total circulation was 10,000. ''Mathematics Magazine'' is a continuation of ''Mathematics News Letter'' (1926-1934) and ''National Mathematics Magazine'' (1934-1945.) Doris Schattschneider became the first female editor of ''Mathematics Magazine'' in 1981. .. The MAA gives the Carl B. Allendoerfer Awards annually "for articles of expository excellence" published in ''Mathematics Magazine''. See also *''American Mathematical Mont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Publications Mathématiques De L'IHÉS
''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique. The journal was established in 1959 and was published at irregular intervals, from one to five volumes a year. It is now biannual. The editor-in-chief is Claire Voisin (Collège de France). See also *''Annals of Mathematics'' *'' Journal of the American Mathematical Society'' *''Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...'' External links * Back issues from 1959 to 2010 Mathematics journals Publications established in 1959 Springer Science+Business Media academic journals Biannual journal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Origami
Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike in traditional origami, the panels of the paper cannot be bent during the folding process; they must remain flat at all times, and the paper only folded along its hinges. A rigid origami model would still be foldable if it was made from glass sheets with hinges in place of its crease lines. However, there is no requirement that the structure start as a single flat sheet – for instance shopping bags with flat bottoms are studied as part of rigid origami. Rigid origami is a part of the study of the mathematics of paper folding, and rigid origami structures can be considered as a type of mechanical linkage. Rigid origami has great practical utility. Mathematics The number of standard origami bases that can be folded using rigid origami is restricted by its rules. Rigid origami does not have to follow the Huzita–Hatori axioms, the fold lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flexagon
In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be ''flexed'' or folded in certain ways to reveal faces besides the two that were originally on the back and front. Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon. In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of ''pats''. Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an equivalence relation. History Discovery and introduction The discovery of the first flexagon, a trihexaflexagon, is credited to the British mathematician Arthur H. Stone, whil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron. Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be ''cut and unfolded'' as nets in 3-space. Definition A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each fa ... [...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] |
Dissection Puzzle
A dissection puzzle, also called a transformation puzzle or ''Richter Puzzle'', is a tiling puzzle where a set of pieces can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles is also considered to be a type of dissection puzzle. Puzzles may include various restraints, such as hinged pieces, pieces that can fold, or pieces that can twist. Creators of new dissection puzzles emphasize using a minimum number of pieces, or creating novel situations, such as ensuring that every piece connects to another with a hinge. History Dissection puzzles are an early form of geometric puzzle. The earliest known descriptions of dissection puzzles are from the time of Plato (427–347 BCE) in Ancient Greece, and involve the challenge of turning two equal squares into one larger square using four pieces. Other ancient dissection puzzles were used as graphic depictions of the Pythagorean theorem (see square trisection). A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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