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Stacked Polytope
In polyhedral combinatorics (a branch of mathematics), a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets.. Examples Every simplex is itself a stacked polytope. In three dimensions, every stacked polytope is a polyhedron with triangular faces, and several of the deltahedra (polyhedra with equilateral triangle faces) are stacked polytopes In a stacked polytope, each newly added simplex is only allowed to touch one of the facets of the previous ones. Thus, for instance, the quadaugmented tetrahedron, a shape formed by gluing together five regular tetrahedra around a common line segment is a stacked polytope (it has a small gap between the first and last tetrahedron). However, the similar-looking pentagonal bipyramid is not a stacked polytope, because if it is formed by gluing tetrahedra together, the last tetrahedron will be glued to two triangular faces of previous tetrahedra instead of only one. Other non-convex stac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cell Net
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's \alpha_4 polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides. The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 matc ... [...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] |
Simplicial Polytope
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's theorem to a maximal planar graph. They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons. Examples Simplicial polyhedra include: * Bipyramids * Gyroelongated dipyramids *Deltahedra (equilateral triangles) ** Platonic *** tetrahedron, octahedron, icosahedron ** Johnson solids: ***triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid * Catalan solids: ** triakis tetrahedron, triakis octahedron, tetrakis hexahedron, disdyakis dodecahedron, triakis icosahedron, pentakis dodecahedron, disdyakis triacontahedron Simplicial tilings: * Regular: ** triangular tiling *Laves tilings: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonian Network
In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction. Definition An Apollonian network may be formed, starting from a single triangle embedded in the Euclidean plane, by repeatedly selecting a triangular face of the embedding, adding a new vertex inside the face, and connecting the new vertex to each vertex of the face containing it. In this way, the triangle containing the new vertex is subdivided into three smaller triangles, which may in turn be subdivided in the same way. Examples The complete graphs on three and four vertices, and , are both Apollonian networks. is formed by starting with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In the mathematical area of graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-tree
In graph theory, a ''k''-tree is an undirected graph formed by starting with a (''k'' + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex ''v'' has exactly ''k'' neighbors ''U'' such that, together, the ''k'' + 1 vertices formed by ''v'' and ''U'' form a clique. Characterizations The ''k''-trees are exactly the maximal graphs with a treewidth of ''k'' ("maximal" means that no more edges can be added without increasing their treewidth). They are also exactly the chordal graphs all of whose maximal cliques are the same size ''k'' + 1 and all of whose minimal clique separators are also all the same size ''k''.. Related graph classes 1-trees are the same as unrooted trees. 2-trees are maximal series–parallel graphs, and include also the maximal outerplanar graphs. Planar 3-trees are also known as Apollonian networks. The graphs that have treewidth at most ''k'' are exactly the subgraphs of ''k''-trees, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undirected Graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
