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Polygons & Tessellations (page 1) 1. To tessellate (or tile) means to cover a flat surface with a repeated pattern, with no gaps or overlaps. Tessellate comes from the Latin tessera, which refers to a small block of stone, tile, or glass used in making a mosaic. 2. Tessellations often consist of a simple figure which is repeatedly translated, rotated, and/or reflected into a pattern. 3. Tessellations occur all around us, in— • nature (seeds, cracks in mud, honeycombs, rock formations) • art (see examples in Geometry textbook p.667) • everyday life (brick patios, tiled floors, wallpaper, tapestries). 4. Because the figures in a tessellation do not overlap or leave gaps, the sum of the measures of the angles around any vertex must be 360°. • If the angles around a vertex are congruent, then the measure of each angle must be a factor of 360. Example 1: Does a regular hexagon tessellate a plane? Answer 1: Step 1. The angle a of each vertex is given by the formula: a = Step 2. Substitute 6 for n: a = = 120 Step 3. Is 120 a factor of 360? Divide 360 by 120: = 3. Step 4. Since 3 is an integer, then 120 is a factor of 360; therefore a regular hexagon will tessellate. Example 2: Does a regular 18-gon tessellate a plane? Answer 2: Step 1. The angle a of each vertex is given by the formula: a = Step 2. Substitute 18 for n: a = = 160 Step 3. Is 160 a factor of 360? Divide 360 by 160: = 2.25. Step 4. Since 2.25 is not an integer, then 160 is not a factor of 360; therefore a regular 18-gon will NOT tessellate. 5. Every triangle tessellates. 6. Every quadrilateral tessellates. Polygons & Tessellations (page 2) (from Prentice-Hall Geometry, p. 669.) dwa 4/07/13