Tessellations all around us

1. Tessellations all around us Look for tessellations in walls, patios and pavements.

2. Tessellations all around us Sometimes an unusual shape will tessellate Common shapes can be arranged in unusual ways

3. Tessellations all around us Sometimes 2 or more different shapes will tessellate.

4. Modern-day Tessellations • Soccer balls • Bathroom floors • Wallpaper designs

5. Examples • Brick walls are tessellations. The rectangular face of each brick is a tile on the wall. • Chess and checkers are played on a tiling. Each colored square on the board is a tile, and the board is an example of a periodic tiling.

6. Alhambra • The Alhambra, a Moor palace in Granada, Spain, is one of today’s finest examples of the mathematical art of 13th century Islamic artists.

7. Regular tiling • Which other regular polygons do you think can tile the plane?

8. Triangles • Triangles? • Yep! • How many triangles to make 1 complete rotation? • The interior angle of every equilateral triangle is 60º. If we sum the angles around a vertex, we get 60º + 60º + 60º + 60º + 60º + 60º = 360º again!.

9. Pentagons • Will pentagons work? • The interior angle of a pentagon is 108º, and 108º + 108º + 108º = 324º.

10. Hexagons • Hexagons? • The interior angle is 120º, and 120º + 120º + 120º = 360º. • How many hexagons to make 1 complete rotation?

11. Not without getting overlaps. In fact, all polygons with more than six sides will overlap. Heptagons • Heptagons? Octagons?

12. Regular tiling • So, the only regular polygons that tessellate the plane are triangles, squares and hexagons. • That was an easy game. Let’s make it a bit more rewarding.

16. Tessellations by M.C. Escher

17. M. C. Escher, Cycle

18. Bulldog (Tessellation 97)

19. Pegasus (Tessellation 105)