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For equilateral zonogons, a -sided one can be tiled by () rhombi.) In this tiling, there is a parallelogram for each pair of slopes of sides in the 2 n {\displaystyle 2n} -sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling. [ 5 ]
The intersection of two convex polygons is a convex polygon. A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Helly's theorem: For every collection of at least three convex polygons: if all intersections of all but one polygon are nonempty ...
Analogously to the two ears theorem, every non-convex simple polygon has at least one mouth. Polygons with the minimum number of principal vertices of both types, two ears and a mouth, are called anthropomorphic polygons. [7] Repeatedly finding and removing a mouth from a non-convex polygon will eventually turn it into the convex hull of the ...
For a simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than a straight angle ...
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.
A related theorem states that every simple polygon that is not a convex polygon has a mouth, a vertex whose two neighbors are the endpoints of a line segment that is otherwise entirely exterior to the polygon. The polygons that have exactly two ears and one mouth are called anthropomorphic polygons. [16]
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The Gauss map of any convex polyhedron maps each face of the polygon to a point on the unit sphere, and maps each edge of the polygon separating a pair of faces to a great circle arc connecting the corresponding two points. In the case of a zonohedron, the edges surrounding each face can be grouped into pairs of parallel edges, and when ...