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Polyhedral Net
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book ''A Course in the Art of Measurement with Compass and Ruler'' (''Unterweysung der Messung mit dem Zyrkel und Rychtscheyd '') included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by Augustin Hirschvogel. Existence and uniqueness Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron Flat
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphs And Combinatorics
''Graphs and Combinatorics'' (ISSN 0911-0119, abbreviated ''Graphs Combin.'') is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. Its editor-in-chief is Katsuhiro Ota of Keio University. The journal was first published in 1985. Its founding editor in chief was Hoon Heng Teh of Singapore, the president of the Southeast Asian Mathematics Society, and its managing editor was Jin Akiyama. Originally, it was subtitled "An Asian Journal". In most years since 1999, it has been ranked as a second-quartile journal in discrete mathematics and theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ... by SCImago Journal Rank.. References {{reflist Publications established in 1985 Combinatorics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tesseract2
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes. The tesseract is also called an 8-cell, C8, (regular) octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a member of the dimensional family of hypercubes or measure polytopes. Coxeter labels it the \gamma_4 polytope. The term ''hypercube'' without a dimension reference is frequently treated as a synonym for this specific polytope. The ''Oxford English Dictionary'' traces the word ''tesseract'' to Charles Howard Hinton's 1888 book ''A New Era of Thought''. The term derives from the Greek ( 'four') and from ( 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Spider And The Fly Problem
400px, Isometric projection and net of naive (1) and optimal (2) solutions of the spider and the fly problem The spider and the fly problem is a recreational geodesics problem with an unintuitive solution. Problem In the typical version of the puzzle, an otherwise empty cuboid room 30 feet long, 12 feet wide and 12 feet high contains a spider and a fly. The spider is 1 foot below the ceiling and horizontally centred on one 12′×12′ wall. The fly is 1 foot above the floor and horizontally centred on the opposite wall. The problem is to find the minimum distance the spider must crawl along the walls, ceiling and/or floor to reach the fly, which remains stationary. Solutions A naive solution is for the spider to remain horizontally centred, and crawl up to the ceiling, across it and down to the fly, giving a distance of 42 feet. The shortest distance strictly abiding by the rules, 40 feet, is obtained by constructing an appropriate net of the room and connecting the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spherical tiling, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun '' geodesic'' and the adjective ''geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blooming (geometry)
In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement of the net in a plane. As in rigid origami, the polygons of the net must remain individually flat throughout the motion, and are not allowed to intersect or cross through each other. A blooming, reversed to go from the flat net to a polyhedron, can be thought of intuitively as a way to fold the polyhedron from a paper net without bending the paper except at its designated creases. An early work on blooming by Biedl, Lubiw, and Sun from 1999 showed that some nets for non-convex but topologically spherical polyhedra have no blooming. The question of whether every convex polyhedron admits a net with a blooming was posed by Robert Connelly, and came to be known as Connelly’s blooming conjecture. More specifically, Miller and Pak suggested in 200 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
