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Császár Polyhedron
In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces. Complete graph The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals: every two vertices of the polygon are connected by an edge, so there is no line segment between two vertices that does not lie on the polyhedron boundary. That is, the vertices and edges of the Császár polyhedron form a complete graph. The combinatorial description of this polyhedron has been described earlier by Möbius. Three additional different polyhedra of this type can be found in a paper by Bokowski, J. and Eggert, A. If the boundary of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Polyhedron
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Variations in definition Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by exactly two polygons, and at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link of the vertex. For toroidal polyhedra, this manifold is an orientable surface. Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus. In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract pol ... [...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] |
Nonconvex Polyhedra
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 5 quasiregular ones, and 48 semiregular ones. There are also two infinite sets of ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have polygon density greater than 1) correspond to circular polygons with overlapping tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhedra with overlapping tiles; there are 47 nonprismatic such uniform star polyhedra. The remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schönhardt Polyhedron
In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who described it in 1928. The same polyhedra have also been studied in connection with Cauchy's rigidity theorem as an example where polyhedra with two different shapes have faces of the same shapes. Construction One way of constructing the Schönhardt polyhedron starts with a triangular prism, with two parallel equilateral triangles as its faces. One of the triangles is rotated around the centerline of the prism, breaking the square faces of the prism into pairs of triangles. If each of these pairs is chosen to be non-convex, the Schönhardt polyhedron is the result. Properties The Schönhardt polyhedron has six vertices, twelve edges, and eight triangular faces. The six vertices of the Schönhardt polyhedron can be used to form fifteen unordered pairs of vertices. Twelv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular polygon, regular hexagon'' has Schläfli symbol and can also be constructed as a Truncation (geometry), truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral polygon, equilateral and equiangular polygon, equiangular. It is bicentric polygon, bicentric, meaning that it is both cyclic polygon, cyclic (has a circumscribed circle) and tangential polygon, tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed figure, inscribed circle). All internal angles are 120 degree (angle), degrees. A regular hexago ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lajos Szilassi
Lajos Szilassi (born 1942 in Szentes, Hungary) was a professor of mathematics at the University of Szeged who worked in projective and non-Euclidean geometry, applying his research to computer generated solutions of geometric problems.Lajos Szilassi is 70 Department of Geometry, Bolyai Institute, Faculty of Science, University of Szeged Biography Szilassi obtained his high school diploma in 1966 at the Bolyai Institute of the , majoring in mathematical representation geometry. He had been teaching for six years in a seco ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice vers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ákos Császár
Ákos Császár ( hu, Császár Ákos, ) (26 February 1924, Budapest – 14 December 2017, Budapest) was a Hungarian mathematician, specializing in general topology and real analysis. He discovered the Császár polyhedron, a nonconvex polyhedron without diagonals. He introduced the notion of syntopogeneous spaces, a generalization of topological spaces. During the end of 1944 his grandfather lost his life during the siege of Budapest. Then his father, older brother and himself were arrested by the Germans and sent to a concentration camp approximatively 45 miles east of Budapest. An infectious illness spread in the camp, and his brother and father died, but Ákos survived. He is a member of the group of five students of the late professor Lipót Fejér who called them "The Big Five". The other four are John Horvath, János Aczél, Steven Gaal and László Fuchs, all of whom are now retired mathematics professors in North America. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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