origami & mathematics mosaics made from triangles, squares

Origami & Mathematics

Mosaics made from triangles, squares and hexagonsAbout interesting geometrical patterns build from simple origami tiles

Krystyna Burczykburczyk@mail.zetosa.com.pl4th International Conference

Didaktik des PapierfaltenFreiburg, 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.

© Krystyna Burczyk

3

Examples of mathematical tilings

© Krystyna Burczyk

4

Square tiles

© Krystyna Burczyk

5

Simple motive made from square tiles – how to

make a tile

© Krystyna Burczyk

6

Simple motive made from square tiles

© Krystyna Burczyk

7

Simple motive made from square tiles

We may change flaps’ positions:

Describe these changes

© Krystyna Burczyk

8

Simple motive made from square tiles

We may change flaps’ positions:

© Krystyna Burczyk

9

Simple motive made from square tiles

Not flatted!

© Krystyna Burczyk

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?

© Krystyna Burczyk

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?

© Krystyna Burczyk

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.

© Krystyna Burczyk

13

Hexagonal tiles

© Krystyna Burczyk

14

Hexagonal tiles – how to make a hexagon tile

© Krystyna Burczyk

15

Hexagonal modified tiles

© Krystyna Burczyk

16

Hexagonal tiling

© Krystyna Burczyk

17

Triangular tiles

© Krystyna Burczyk

18

Triangular tiles – how to make a triangle tile

© Krystyna Burczyk

19

Triangular tiling

© Krystyna Burczyk

20

Tilings from hexagonal,

triangular and square tiles

© Krystyna Burczyk

21

Simple tilings from hexagonal, triangular and square tiles

- examples

© Krystyna Burczyk

22

Tilings from hexagonal, triangular and square tiles - examples

© Krystyna Burczyk

23

Tilings from hexagonal, triangular and square tiles

© Krystyna Burczyk

24

Dodecagon made from 1 hexagon, 6 squares and 6 triangles

© Krystyna Burczyk

25

Dodecagon made from 1 hexagon, 6 squares and 6 triangles

© Krystyna Burczyk

26

Dodecagon made from one hexagon, six squares and six

triangles

© Krystyna Burczyk

27

Tilings from dodecagonal,

triangular and square tiles

© Krystyna Burczyk

28

Tilings from dodecagons, squares and triangles

© Krystyna Burczyk

29

How to prepare paper for

mosaic motives?

© Krystyna Burczyk

30

How prepare the paper for tiling from hexagons, squares

and triangles

© Krystyna Burczyk

31

Multi-tiles

© Krystyna Burczyk

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

© Krystyna Burczyk

33

How to pack the plane with multi-tiles?

© Krystyna Burczyk

34

How to pack the plane with that kind of tiles?

© Krystyna Burczyk

35

How to pack the plane with that kind of tiles

© Krystyna Burczyk

36

Fractal structures

© Krystyna Burczyk

37

Sequence of multi-tiles

Figure 0 = SquareFigure 1 = Multi-tile from 5 squaresFigure 2 = Multi-tile made from 5 figures 1Figure 3 = Multi-tile made from 5 figures 2

Continue …

© Krystyna Burczyk

38

Sequence of multi-tiles

Figure 0 = HexagonFigure 1 = Multi-tile from 7 hexagonsFigure 2 = Multi-tile made from 7 figures 1Figure 3 = Multi-tile made from 7 figures 2

Continue …

© Krystyna Burczyk

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!

© Krystyna Burczyk

40

Bibliography

• http://www.uwgb.edu/dutchs/symmetry/uniftil.htm

• http://en.wikipedia.org/wiki/Tiling_by_regular_polygons

© Krystyna Burczyk