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Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points , lines , planes , circles , spheres , polygons , and so forth.
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.
In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. [1] That is, it is a polytope that equals the convex hull of its integer points. [2] Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional ...
In elementary geometry, a polytope is a geometric object with flat sides . Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope.
Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum [1] and Ziegler [2] on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out ...
Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry.
In geometry, more specifically in polytope theory, Kalai's 3 d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. [1] It states that every d-dimensional centrally symmetric polytope has at least 3 d nonempty faces (including the polytope itself as a face but not including the ...
Net. In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C 120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron [1] and hecatonicosahedroid.
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