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§8.1 Polygons The student will learn: the definition of polygons, The terms associated with polygons, and how to tessellate a surface. 1 Polygon, Convexity Definitions A polygon is a closed plane figure formed by three or more line segments. A convex polygon is a polygon in which all of the interior angles are less than 180. A concave polygon has at least one interior angles that is greater than 180. 2 More Terms Definitions An side is one of the segments forming the polygon. An interior angle is formed by the intersection of two adjacent sides. A exterior angle is formed by one of the sides and an adjacent side extended. 3 More Terms Definitions An equilateral polygon is one in which all sides are equal. An equiangular polygon is one in which all interior angles are equal. A regular polygon is both equilateral and equiangular. An diagonal is a segment joining two nonadjacent vertices. 4 And Even More Terms Polygons are named according to their number of sides. # Sides Name 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 12 dodecagon 5 Theorem The sum of the interior angles of a polygon of n sides is 180 (n – 2). Proof left for homework. 6 Theorem In a regular polygon of n sides each interior angle has a measure of: 180(n 2) n Proof left for homework. 7 Theorem The exterior angle of a regular polygon of n sides has a measure of: 360 n Proof left for homework. 8 Theorem The sum of the exterior angles of a polygon is 360. Proof left for homework. 9 Tessellations with Regular Polygons Tessellation Definition A tiling or tessellation of the plane is a collection of regions T 1, T 2, . . . , T n, called tiles such that 1. no two tiles have any interior points in common, and 2. the collection of tiles completely covers the plane. 11 Terms Definitions A tiling that uses only one shape is called a monohedral tessellation. A tiling that uses congruent regular polyhedron is called a regular tessellation. You should be able to prove (One of my favorite final questions.) that there are only three regular tessellations. 12 Semiregular Tessellations There is a way to classify tessellations by choosing a vertex and listing in sequence the number of sides of each regular polygon surrounding that vertex. A semiregular tessellation is a tessellation that is made up of regular polyhedron only and have the same combination of polyhedron at each vertex. There are eight semiregular tessellations. 13 Examples This is a 4.8.8 tessellation. This is a 3.6.3.6 tessellation. The three regular and eight semiregular tessellations are called Archimedean tilings. 14 Uniform Tessellations The Archimedean tilings are also called 1uniform tilings because all the vertices in a tiling are identical. A 2-uniform tiling has two different types of vertices and a 3-uniform has three. 15 Examples 2-uniform tiling 3,6,3,6 or 3,4,4,6 3-uniform tiling 4,4,4,4 or 3,3,4,3,4 or 3,3,3,4,4 16 Examples 2-uniform tiling 3,6,3,6 or 3,4,4,6 3-uniform tiling 4,4,4,4 or 3,3,4,3,4 or 3,3,3,4,4 17 Note There are 20 different 2-uniform tessellations. There are 61 different 3-uniform tessellations. The number of different 4-uniform tessellations is still an unsolved problem. 18 Tessellations with Nonregular Polygons Triangles Any triangle may be used to form a monohedral tessellation. 20 Quadrilaterals Any quadrilateral may be used to create a monohedral tessellation. Bricks, ceiling tiles and all others. 21 Pentagons A regular pentagon does not tessellate, but there are some nonregular that do. Cairo tiling Some others 22 Other Tessellations The literature on tessellations is very extensive. Indeed geometric translations, reflections and rotations may be used along with other geometric techniques. It is a fun and rewarding topic. 23 Tessellation Rocks Tasmania, Under, Down Under Assignment: §8.1

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