DEFINITION: TILES AND TILING 13.3 DEFINITION: TILES AND TILING 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. Copyright © 2008 Pearson Education, Inc.

TILING WITH REGULAR POLYGONS 13.3 TILING WITH REGULAR POLYGONS 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. Copyright © 2008 Pearson Education, Inc.

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

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

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

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

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

TILING THE PLANE WITH CONGRUENT POLYGONAL TILES 13.3 TILING THE PLANE WITH CONGRUENT POLYGONAL TILES 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). Copyright © 2008 Pearson Education, Inc.

SEMIREGULAR TILINGS OF THE PLANE 13.3 SEMIREGULAR TILINGS OF THE PLANE 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. Copyright © 2008 Pearson Education, Inc.

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

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