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The most common version uses the concept of "equidecomposability" of polygons: two polygons are equidecomposable if they can be split into finitely many triangles that only differ by some isometry (in fact only by a combination of a translation and a rotation). In this case the Wallace–Bolyai–Gerwien theorem states that two polygons are ...
The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...
A polygon ear. One way to triangulate a simple polygon is based on the two ears theorem, as the fact that any simple polygon with at least 4 vertices without holes has at least two "ears", which are triangles with two sides being the edges of the polygon and the third one completely inside it. [5]
A compound of two "line segment" digons, as the two possible alternations of a square (note the vertex arrangement). The apeirogonal hosohedron , containing infinitely narrow digons. Any straight-sided digon is regular even though it is degenerate, because its two edges are the same length and its two angles are equal (both being zero degrees).
Convex equilateral pentagon dissected into 3 triangles, which helps to calculate the value of angle δ as a function of α and β. When a convex equilateral pentagon is dissected into triangles, two of them appear as isosceles (triangles in orange and blue) while the other one is more general (triangle in green).
It is always possible to partition a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polygons as possible is described by Chazelle & Dobkin (1985). [5] A triangle can never be concave, but there exist concave polygons with n sides for any n > 3.
It is possible to divide an equilateral triangle into three congruent non-convex pentagons, meeting at the center of the triangle, and to tile the plane with the resulting three-pentagon unit. [21] A similar method can be used to subdivide squares into four congruent non-convex pentagons, or regular hexagons into six congruent non-convex ...
One of the two uses the symmetry axis as one of the two perpendicular lines, while the other has two lines of slope , each crossing the base and one side. [ 1 ] This subdivision of a triangle is a special case of a theorem of Richard Courant and Herbert Robbins that any plane area can be subdivided into four equal parts by two perpendicular ...