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Tessellations 7/30/2009 Title slide © Powered Chalk LLC 2009 Definition A tessellation is a filling up of a 2-dimensional space with the same shape without any overlap or spaces. A tessellation of hexagons A tessellation of triangles. A tessellation of trapezoids © Powered Chalk LLC 2009 Empty spaces, overlap Tessellations cannot have any empty spaces or overlap. Overlap Open spaces Pentagons do not tessellate. © Powered Chalk LLC 2009 © Powered Chalk LLC 2009 1 Tessellations 7/30/2009 Sum of corner angles is 360º Shapes that tessellate join at the vertices to add up 360⁰ © Powered Chalk LLC 2009 Fraction of 360º Shapes that don’t tessellate have a fractional part missing from 360⁰ Does not fill the circle or 360º. © Powered Chalk LLC 2009 Mathematical Concepts The measures of the angles that meet at the corners must add up to exactly 360º. The sum of all the angles in a polygon = ( n - 2 )·180º, where n is the number of sides. The measure of one angle is the sum of the angles divided by n. For equal numbers to add up to 360º they must go into 360º evenly. © Powered Chalk LLC 2009 © Powered Chalk LLC 2009 2 Tessellations 7/30/2009 Ex. 1 Does an octagon tessellate? Find the sum of the measures of all the interior angles. Sum m’s = 180 (n-2) =180 (8-2) =180 (6) Sum m’s =1080º Find the measure of one angle. m = 1080 8 space m = 135º Divide 360 ⁰ by 135º. Ratio = 360/135= 2.67 Conclusion: An octagon does not tessellate because the ratio of 360 to 135 is not a whole number. overlap © Powered Chalk LLC 2009 Ex. 2 Does a hexagon tessellate? Find the sum of the measures of all the interior angles. Sum m’s = 180 (n-2) =180 (6-2) =180 (4) Sum m’s =720º Find the measure of one angle. m = 720 6 m = 120º Divide 360 ⁰ by 120º. Ratio = 360/120= 3 Conclusion: An octagon does tessellate because 120 goes evenly into 360. © Powered Chalk LLC 2009 Websites of Interest Checked on 7/11/09 Math Forum: Tessellation Tutorials by Suzanne Alejandre Shodor Interactivate: Tessellate! Your feedback is important to us, send us your comments. www.basicswithpower.com © Powered Chalk LLC 2009 © Powered Chalk LLC 2009 3