In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions,
representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets.
The same calculation can be performed for any convex polyhedron, even one without symmetries, by choosing any point interior to the polyhedron as its center. For these polyhedra, the density will be 1.
More generally, for any non-self-intersecting (acoptic) polyhedron, the density can be computed as 1 by a similar calculation that chooses a ray from an interior point that only passes through facets of the polyhedron, adds one when this ray passes from the interior to the exterior of the polyhedron, and subtracts one when this ray passes from the exterior to the interior of the polyhedron. However, this assignment of signs to crossings does not generally apply to star polyhedra, as they do not have a well-defined interior and exterior.
Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point.
The density of a polygon is the number of times that the polygonal boundary winds around its center. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem.
The density of a polygon can also be called its turning number; the sum of the turn angles of all the vertices divided by 360°. This will be an integer for all unicursal paths in a plane.
The density of a compound polygon is the sum of the densities of the component polygons.
For a regular star polygon {p/q}, the density is q.
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In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures.
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: Polyhedra which self-intersect in a repetitive way. Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains. Mathematical studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals of the uniform 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.
We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the g ...
Let P be a d-dimensional convex polytope with n facets F1,F2,....,Fn. The (combinatorial) representation of a face F of P is the set of facet indices j such that F c Fj. Given the representations of all vertices of P, the combinatorial face enumeration pro ...
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In this paper, the flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions. The current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of m ...