1. Tessellations Unit 2 – Congruence

2. Tiling (tessellations) Partition of the infinite plane into pieces having a finite number of distinct shapes. The pieces have to be 2-dimensional and must not overlap in more than their boundaries. Two types: periodic and aperiodic, depending on whether they have any translational symmetries. Geometry Institute2

3. What we should know….. In general, a plane tessellation is a pattern of one or more shapes, completely covering the plane without any gaps or overlaps. Regular Tessellations Semi-Regular Tessellations 3

4. Periodic Tilings Regular Semi-Regular Non-Regular Escher Geometry Institute4

5. Regular Tilings: Pieces are regular polygons Triangle: Geometry Institute5

6. Regular Tilings: Pieces are regular polygons Square: Geometry Institute6

7. Regular Tilings: Pieces are regular polygons Pentagon: Geometry Institute7

8. Regular Tilings: Pieces are regular polygons Hexagon: Geometry Institute8

9. Regular Tilings: Pieces are regular polygons Pentagon: Geometry Institute9

10. Regular Tilings Geometry Institute10

11. Semi-regular Tilings Also called Archimedean tessellations Two or more convex regular polygons Vertex rule : –Same polygons in the same order surround each polygon vertex Geometry Institute11

12. Vertex Rule 3.6.3.63.3.3.3.6 Geometry Institute12

13. It breaks the vertex rule! Do you see where? Not a semi-regular tiling Geometry Institute13

14. Semi-regular Tilings 1) 3.3.3.3.6 2) 3.3.3.4.4 3) 3.3.4.3.4 4) 3.4.6.4 5) 3.6.3.6 6) 3.12.12 7) 4.6.12 8) 4.8.8 Geometry Institute14

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17. Example 3.3.4.3.4 Geometry Institute17

18. Geometry Institute Investigation Will any quadrilateral tessellate? 18

19. Geometry Institute Investigation Will any quadrilateral tessellate? A B C D 19

20. Non-regular tilings Geometry Institute20

21. Geometry Institute M.C.E. Name the shape that is tessellating in each of the following images: 21

22. Geometry Institute M.C.E. M.C. Escher (1898-1972) Master of creating non-polygonal shapes which tessellation… But how?… Movies 22

23. Geometry Institute M.C.E. Objectives I. Discover the underlying polygonal grid M.C. Escher used to create his work. II. Discuss the symmetry group for each tessellation. III.Explore the tile formation process and it’s relationship to the set of isometries for each tessellation. 23

24. Geometry Institute M.C.E. Pegasus 24

25. Geometry Institute M.C.E. For the tessellation P, we write S(P) for the set of isometries, f, of the plane such that the image of every tile of P under f is a tile of P. S(P) is a group, called a symmetry group for the tessellation. Reading Comprehension…… If all of the tiles of a tessellation can be obtained as isometries of a single tile, we say that the tessellation is monohedral. 25

26. Geometry Institute M.C.E. Dogs 26

27. Geometry Institute M.C.E. Lizards 27

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37. Aperiodic Tiling No translational symmetry Patterns do not repeat throughout the plane in the same pattern. Geometry Institute37

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39. Aperiodic Tilings - Penrose Geometry Institute39

40. Geometry Institute M.C.E. Thank You!!! 40