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An example of a convex polygon: a regular pentagon. In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). [1]
A regular polygon is a zonogon if and only if it has an even number of sides. [2] Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the rectangles , the rhombi , and the parallelograms .
Convex regular icosahedron. Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices. Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion. Then P and Q are themselves congruent.
Then as Tutte showed, this system of equations will have a unique solution in which each face of the graph is drawn as a convex polygon. [17] Intuitively, this solution describes the pattern that would be obtained by replacing the interior edges of the graph by ideal springs and letting them settle to their minimum-energy state. [18]
A polygon is called sweepable, if a straight line may be continuously moved over the whole polygon in such a way that at any moment its intersection with the polygonal area is a convex set. A monotone polygon is sweepable by a line which does not change its orientation during the sweep.
Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges. There are seven families of isogonal figures , each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the ...
If you’re stuck on today’s Wordle answer, we’re here to help—but beware of spoilers for Wordle 1304 ahead. Let's start with a few hints.
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, [1] whereas mathematical optimization is in general NP-hard. [2 ...
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