Copyright © 2008 Pearson Education, Inc. Slide 13-25

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.

Copyright © 2008 Pearson Education, Inc. Slide 13-26

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.

Copyright © 2008 Pearson Education, Inc. Slide 13-27

TILING WITH EQUILATERAL TRIANGLES

13.3

( )6 60 360° = °

One interior angle of an equilateral triangle has measure 60.

At a vertex angle:

Copyright © 2008 Pearson Education, Inc. Slide 13-28

TILING WITH SQUARES

13.3

( )4 90 360° = °

One interior angle of a square has measure 90.

At a vertex angle:

Copyright © 2008 Pearson Education, Inc. Slide 13-29

TILING WITH REGULAR HEXAGONS

13.3

( )3 120 360° = °

One interior angle of a regular hexagon has measure

At a vertex angle:

( )(6 2) 180 720120 .

6 6

- ° °= = °

Copyright © 2008 Pearson Education, Inc. Slide 13-30

TILING WITH REGULAR PENTAGONS?

13.3

( )(5 2) 180108

5

- °= °

One interior angle of a regular pentagon has measure

At a vertex angle:

( )3 108 324

leaves a gap.

° = ° ( )4 108 432

overlaps.

° = °

Copyright © 2008 Pearson Education, Inc. Slide 13-31

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.

Copyright © 2008 Pearson Education, Inc. Slide 13-32

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).

Copyright © 2008 Pearson Education, Inc. Slide 13-33

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.

Copyright © 2008 Pearson Education, Inc. Slide 13-34

TILINGS OF ESCHER TYPE

13.3

Dutch artist Escher created a large number of artistic tilings.

ESCHER’S BIRDS ITS GRID OF PARALLELOGRAMS

Copyright © 2008 Pearson Education, Inc. Slide 13-35

TILINGS OF ESCHER TYPE

13.3

MODIFYING A REGULAR HEXAGON

WITH ROTATIONS

CREATES: