Tessellations All Around Us
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Tessellations All Around Us Presentation Transcript

  • 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.
    • 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.
  • 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.
  • Regular tiling
    • Which other regular polygons do you think can tile the plane?
  • Triangles
    • Triangles?
    • 7. Yep!
    • 8. How many triangles to make 1 complete rotation?
    • 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!.
  • Pentagons
    • Will pentagons work?
    • 10. The interior angle of a pentagon is 108º, and 108º + 108º + 108º = 324º.
  • Hexagons
    • Hexagons?
    • 11. The interior angle is 120º, and 120º + 120º + 120º = 360º.
    • 12. How many hexagons to make 1 complete rotation?
    • Not without getting overlaps. In fact, all polygons with more than six sides will overlap.
    Heptagons
    • Heptagons? Octagons?
  • Regular tiling
    • So, the only regular polygons that tessellate the plane are triangles, squares and hexagons.
    • 13. That was an easy game. Let’s make it a bit more rewarding.
  • 14.
  • 15.
  • 16. Tessellations by M.C. Escher
  • 17. M. C. Escher, Cycle
  • 18. Bulldog
    (Tessellation 97)
  • 19. Pegasus
    (Tessellation 105)