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Common Net
In geometry, a common net is a Net (polyhedron), net that can be folded onto several Polyhedron, polyhedra. To be a valid common net, there should not exist any non-overlapping sides, and the resulting polyhedra must be connected through faces. Examples of these particular nets in the research date back to the end of the 20th century; despite that, not many examples have been found. Two classes, however, have been deeply explored, regular polyhedra and cuboids. The search for common nets is usually made by either an extensive search or the overlapping of nets that tile the plane. proved that every convex polyhedron can be unfolded and refolded to a different convex polyhedron. There are types of common nets: strict edge unfoldings and free unfoldings. Strict edge unfoldings refer to common nets where the different polyhedra that can be folded use the same folds: to fold one polyhedra from the net of another, there is no need to make new folds. Free unfoldings refer to the opposit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (polyhedron)
In geometry, a net of a polyhedron is an arrangement of non-overlapping Edge (geometry), edge-joined polygons in the plane (geometry), plane which can be folded (along edges) to become the face (geometry), 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 poly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Folding Algorithms
''Geometric Folding Algorithms: Linkages, Origami, Polyhedra'' is a monograph on the mathematics and computational geometry of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by Cambridge University Press (). A Japanese-language translation by Ryuhei Uehara was published in 2009 by the Modern Science Company (). Audience Although aimed at computer science and mathematics students, much of the book is accessible to a broader audience of mathematically-sophisticated readers with some background in high-school level geometry. Mathematical origami expert Tom Hull has called it "a must-read for anyone interested in the field of computational origami". It is a monograph rather than a textbook, and in particular does not include sets of exercises. The Basic Library List Committee of the Mathematical Association of America has recommended this book for inclusion in undergraduate mathematics libraries. Topics and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Square Bipyramid
In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramid, equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid. The dual polyhedron of the gyroelongated square bipyramid is a square truncated trapezohedron with eight pentagons and two squares as its faces. The gyroelongated square pyramid appears in chemistry as the basis for the bicapped square antiprismatic molecular geometry, and in mathematical optimization as a solution to the Thomson problem. Construction Like other Gyroelongated bipyramid, gyroelongated bipyramids, the gyroelongated square bipyramid can be constructed by attaching two Equilateral square pyramid, equilateral sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Disphenoid
In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its face (geometry), faces. It is an example of deltahedron and Johnson solid. It can be constructed in different approaches. This shape is also called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron. The snub disphenoid can be visualized as an atom cluster surrounding a central atom, that is the dodecahedral molecular geometry. Its vertices may be placed in a sphere and can also be used as a minimum possible Lennard-Jones potential among all eight-sphere clusters. The dual polyhedron of the snub disphenoid is the elongated gyrobifastigium. Construction The snub disphenoid can be constructed in different ways. As suggested by the name, the snub disphenoid is constructed from a tetragonal disphenoid by cutting all the edges from its faces, and adding equilateral triangles (the light blue colors in the following image) that are twisted in a certain a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex set, convex and non-convex shapes. Combinatorially equivalent to the regular octahedron The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: * Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuboid Common Net
In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six faces; it has eight vertices and twelve edges. A ''rectangular cuboid'' (sometimes also called a "cuboid") has all right angles and equal opposite rectangular faces. Etymologically, "cuboid" means "like a cube", in the sense of a convex solid which can be transformed into a cube (by adjusting the lengths of its edges and the angles between its adjacent faces). A cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube. General cuboids have many different types. When all of the rectangular cuboid's edges are equal in length, it results in a cube, with six square faces and adjacent faces meeting at right angles. Along with the rectangular cuboids, ''parallelepiped'' is a cuboid with six parallelogram faces. ''Rhombohedron'' is a cuboid with six rhombus faces. A '' square frustum'' is a frustum with a square base, but the rest of its faces are quadrilaterals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
