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Copyright © 2008 Pearson Education, Inc. Slide DEFINITION: TILES AND TILING 13.3 A simple closed curve, together with its interior, is a tile. A set of tiles forms a tiling of a figure if the figure is completely covered by the tiles without overlapping any interior points of the tiles. In a tiling of a figure, there can be no gaps between tiles. Tilings are also known as tessellations.

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Copyright © 2008 Pearson Education, Inc. Slide TILING WITH REGULAR POLYGONS 13.3 Any arrangement of nonoverlapping polygonal tiles surrounding a common vertex is called a vertex figure. Equilateral triangles form a regular tiling because the measures of the interior angles meeting at a vertex figure add to 360.

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Copyright © 2008 Pearson Education, Inc. Slide TILING WITH EQUILATERAL TRIANGLES 13.3 One interior angle of an equilateral triangle has measure 60. At a vertex angle:

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Copyright © 2008 Pearson Education, Inc. Slide TILING WITH SQUARES 13.3 One interior angle of a square has measure 90. At a vertex angle:

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Copyright © 2008 Pearson Education, Inc. Slide TILING WITH REGULAR HEXAGONS 13.3 One interior angle of a regular hexagon has measure At a vertex angle:

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Copyright © 2008 Pearson Education, Inc. Slide TILING WITH REGULAR PENTAGONS? 13.3 One interior angle of a regular pentagon has measure At a vertex angle:

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Copyright © 2008 Pearson Education, Inc. Slide THE REGULAR TILINGS OF THE PLANE 13.3 There are exactly three regular tilings of the plane: by equilateral triangles, by squares, and by regular hexagons.

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Copyright © 2008 Pearson Education, Inc. Slide TILING THE PLANE WITH CONGRUENT POLYGONAL TILES 13.3 The plane can be tiled by: any triangular tile; any quadrilateral tile, convex or not; certain pentagonal tiles (for example, those with two parallel sides); certain hexagonal tiles (for example, those with two opposite parallel sides of the same length).

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Copyright © 2008 Pearson Education, Inc. Slide SEMIREGULAR TILINGS OF THE PLANE 13.3 An edge-to-edge tiling of the plane with more than one type of regular polygon and with identical vertex figures is called a semiregular tiling.

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Copyright © 2008 Pearson Education, Inc. Slide TILINGS OF ESCHER TYPE 13.3 Dutch artist Escher created a large number of artistic tilings. ESCHERS BIRDS ITS GRID OF PARALLELOGRAMS

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Copyright © 2008 Pearson Education, Inc. Slide TILINGS OF ESCHER TYPE 13.3 MODIFYING A REGULAR HEXAGON WITH ROTATIONS CREATES:

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