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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 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 ...
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 ...
One pair of opposite sides is parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent triangles. The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
A non-convex regular polygon is a regular star polygon. The most common example is the pentagram , which has the same vertices as a pentagon , but connects alternating vertices. For an n -sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as { n / m }.
They form the interiors of bounded convex polygons or unbounded convex regions. These are the connected components of the points that would remain after removing all points on lines. [1] The edges or panels of the arrangement are one-dimensional regions belonging to a single line. They are the open line segments and open infinite rays into ...
Corollary: every rectilinear polygon has at least 4 convex corners. The number of knobs (sides connecting two convex corners) is four more than the number of antiknobs (sides connecting two concave corners).To see why, let X be the number of convex corners and Y the number of concave corners. By the previous fact, X=Y+4.
In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted). [ 1 ] [ 2 ] Parallelogons have an even number of sides and opposite sides that are equal in length.