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Tessellations

Tessellations. Tessellation. A tessellation or a tiling is a way to cover a floor with shapes so that there is no overlapping or gaps. Remember the last jigsaw puzzle piece you put together? Well, that was a tessellation. The shapes were just really weird. Examples.

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Tessellations

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  1. Tessellations

  2. Tessellation • A tessellation or a tiling is a way to cover a floor with shapes so that there is no overlapping or gaps. • Remember the last jigsaw puzzle piece you put together? Well, that was a tessellation. The shapes were just really weird.

  3. 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.

  4. Examples • Mother nature is a great producer of tilings. The honeycomb of a beehive is a periodic tiling by hexagons. • Each piece of dried mud in a mudflat is a tile. This tiling doesn't have a regular, repeating pattern. Every tile has a different shape. In contrast, in our other examples there was just one shape.

  5. 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.

  6. Tesselmania • Motivated by what he experienced at Alhambra, Maurits Cornelis Escher created many tilings.

  7. Regular tiling • To talk about the differences and similarities of tilings it comes in handy to know some of the terminology and rules. • We’ll start with the simplest type of tiling, called a regular tiling.It has three rules: The tessellation must cover a plane with no gaps or overlaps. The tiles must be copies of one regular polygon. Each vertex must join another vertex. • Can we tessellate using these game rules? Let’s see.

  8. Regular tiling • Tessellations with squares, the regular quadrilateral, can obviously tile a plane. • Note what happens at each vertex. The interior angle of each square is 90º. If we sum the angles around a vertex, we get 90º + 90º + 90º + 90º = 360º. • How many squares to make 1 complete rotation?

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

  10. 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!.

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

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

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

  14. 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.

  15. Here is an example made from a square, hexagon, and dodecahedron: Semiregular tiling • A semiregular tiling has the same game rules except that now we can use more than one type of regular polygon. • To name a tessellation, work your way around one vertex counting the number of sides of the polygons that form the vertex. • Go around the vertex such that the smallest possible numbers appear first.

  16. Semiregular tiling • Here is another example made from three triangles and two squares: • There are only 8 semiregular tessellations, and we’ve now seen two of them: the 4.6.12 and the 3.3.4.3.4 • Your in-class construction will help you find the remaining 6 semiregular tessellations.

  17. Demiregular tiling • The 3 regular tessellations (by equilateral triangles, by squares, and by regular hexagons) and the 8 semiregular tessellations you just found are called 1-uniform tilings because all the vertices are identical. • If the arrangement at each vertex in a tessellation of regular polygons is not the same, then the tessellation is called a demiregular tessellation. • If there are two different types of vertices, the tiling is called 2-uniform. If there are three different types of vertices, the tiling is called 3-uniform.

  18. Examples • There are 20 different 2-uniform tessellations of regular polygons. 3.4.6.4 / 4.6.12 3.3.3.3.3.3 / 3.3.3.4.4 / 3.3.4.3.4

  19. Summary • Regular Tessellation • Only one regular polygon used to tile • Semiregular Tessellation • Uses more than one regular polygon • Has the same pattern of polygons AT EVERY VERTEX • Demiregular Tessellation • Uses more than one regular polygon • Has DIFFERENT patterns of polygons used at vertices • Must name all different patterns.

  20. Name the Tessellation Regular? SemiRegular? DemiRegular? SemiRegular 4.6.12

  21. Name the Tessellation Regular? SemiRegular? DemiRegular? Demiregular 3.12.12/3.4.3.12

  22. Name the Tessellation Regular? SemiRegular? DemiRegular? Demiregular 3.3.3.3.3.3/3.3.4.12

  23. Name the Tessellation Regular? SemiRegular? DemiRegular? DemiRegular 3.6.3.6/3.3.6.6

  24. Name the Tessellation Regular? SemiRegular? DemiRegular? SemiRegular 3.3.4.3.4

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