Origami & Mathematics Mosaics made from triangles, squares and hexagons About interesting geometrical patterns build from simple origami tiles

Transcription

1 Origami & Mathematics Mosaics made from triangles, squares and hexagons About interesting geometrical patterns build from simple origami tiles Krystyna Burczyk 4th International Conference Didaktik des Papierfalten Freiburg, November 2009

2 What does mean a tiling (tesselation)? A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. 2

3 Examples of mathematical tilings 3

4 Square tiles 4

5 Simple motive made from square tiles how to make a tile 5

6 Simple motive made from square tiles 6

7 Simple motive made from square tiles We may change flaps positions: Describe these changes 7

8 Simple motive made from square tiles We may change flaps positions: 8

9 Simple motive made from square tiles Not flatted! 9

10 Simple motive made from square tiles how to compose pattern from 4 tiles Take a square mosaic 2x2 made from 4 tiles: 2 in one color, 2 in the second color. Change flaps positions. What kind of symmetry patterns can you obtain? How many possibilities exist? 10

11 Simple motive made from square tiles how to compose pattern from 9 tiles Take a square mosaic 3x3 made from duo-color paper. Change flap positions. What kind of symmetry patterns can you obtain? How many posibilities exist? 11

12 A few problems - the simple square motives 1. There is a 9x9 mosaic from gray-white paper. How many white motives may we put in this mosaic (motives shouldn t have common points)? 2. Find interesting symmetric patterns in the 9x9 mosaic. 12

13 Hexagonal tiles 13

14 Hexagonal tiles how to make a hexagon tile 14

15 Hexagonal modified tiles 15

16 Hexagonal tiling 16

17 Triangular tiles 17

18 Triangular tiles how to make a triangle tile 18

19 Triangular tiling 19

20 Tilings from hexagonal, triangular and square tiles 20

21 Simple tilings from hexagonal, triangular and square tiles - examples 21

22 Tilings from hexagonal, triangular and square tiles - examples 22

23 Tilings from hexagonal, triangular and square tiles 23

24 Dodecagon made from 1 hexagon, 6 squares and 6 triangles 24

25 Dodecagon made from 1 hexagon, 6 squares and 6 triangles 25

26 Dodecagon made from one hexagon, six squares and six triangles 26

27 Tilings from dodecagonal, triangular and square tiles 27

28 Tilings from dodecagons, squares and triangles 28

29 How to prepare paper for mosaic motives? 29

30 How prepare the paper for tiling from hexagons, squares and triangles 30

31 Multi-tiles 31

32 Multi-tiles Multi-tiles = tiles made from a few the same shape and size tiles. Example. Mosaic from multi-tiles made from 4 square tiles 32

33 How to pack the plane with multi-tiles? 33

34 How to pack the plane with that kind of tiles? 34

35 How to pack the plane with that kind of tiles 35

36 Fractal structures 36

37 Sequence of multi-tiles Figure 0 = Square Figure 1 = Multi-tile from 5 squares Figure 2 = Multi-tile made from 5 figures 1 Figure 3 = Multi-tile made from 5 figures 2 Continue 37

38 Sequence of multi-tiles Figure 0 = Hexagon Figure 1 = Multi-tile from 7 hexagons Figure 2 = Multi-tile made from 7 figures 1 Figure 3 = Multi-tile made from 7 figures 2 Continue 38

39 Discovering Mathematics Through Origami Don t be afraid to use glue! A flap is better than flat! Be brave = be creative! Find as many as possible! 39

40 Bibliography