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# Tessellations All Around Us

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### Tessellations All Around Us

1. 1. Tessellations all around us<br />Look for tessellations in walls, patios and pavements.<br />
2. 2. Tessellations all around us<br />Sometimes an unusual shape will tessellate<br />Common shapes can be arranged in unusual ways<br />
3. 3. Tessellations all around us<br />Sometimes 2 or more different shapes will tessellate.<br />
4. 4. Modern-day Tessellations<br />Soccer balls<br />Bathroom floors<br />Wallpaper designs<br />
5. 5. Examples<br /><ul><li>Brick walls are tessellations. The rectangular face of each brick is a tile on the wall.
6. 6. 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.</li></li></ul><li>Alhambra<br /><ul><li>The Alhambra, a Moor palace in Granada, Spain, is one of today’s finest examples of the mathematical art of 13th century Islamic artists.</li></li></ul><li>Regular tiling<br /><ul><li>Which other regular polygons do you think can tile the plane?</li></li></ul><li>Triangles<br /><ul><li>Triangles?
7. 7. Yep!
8. 8. How many triangles to make 1 complete rotation?
9. 9. 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!.</li></li></ul><li>Pentagons<br /><ul><li>Will pentagons work?
10. 10. The interior angle of a pentagon is 108º, and 108º + 108º + 108º = 324º.</li></li></ul><li>Hexagons<br /><ul><li>Hexagons?
11. 11. The interior angle is 120º, and 120º + 120º + 120º = 360º.
12. 12. How many hexagons to make 1 complete rotation?</li></li></ul><li><ul><li>Not without getting overlaps. In fact, all polygons with more than six sides will overlap.</li></ul>Heptagons<br /><ul><li>Heptagons? Octagons?</li></li></ul><li>Regular tiling<br /><ul><li>So, the only regular polygons that tessellate the plane are triangles, squares and hexagons.
13. 13. That was an easy game. Let’s make it a bit more rewarding.</li></li></ul><li>
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16. 16. Tessellations by M.C. Escher<br />
17. 17. M. C. Escher, Cycle<br />
18. 18. Bulldog<br />(Tessellation 97)<br />
19. 19. Pegasus<br />(Tessellation 105)<br />