Origami & Mathematics Mosaics made from triangles, squares ...
Transcript of 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 [email protected] International Conference
Didaktik des PapierfaltenFreiburg, November 2009
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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.
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Examples of mathematical tilings
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Square tiles
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Simple motive made from square tiles – how to
make a tile
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Simple motive made from square tiles
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Simple motive made from square tiles
We may change flaps’ positions:
Describe these changes
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Simple motive made from square tiles
We may change flaps’ positions:
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Simple motive made from square tiles
Not flatted!
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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?
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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?
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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.
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Hexagonal tiles
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Hexagonal tiles – how to make a hexagon tile
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Hexagonal modified tiles
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Hexagonal tiling
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Triangular tiles
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Triangular tiles – how to make a triangle tile
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Triangular tiling
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Tilings from hexagonal,
triangular and square tiles
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Simple tilings from hexagonal, triangular and square tiles
- examples
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Tilings from hexagonal, triangular and square tiles - examples
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Tilings from hexagonal, triangular and square tiles
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Dodecagon made from 1 hexagon, 6 squares and 6 triangles
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Dodecagon made from 1 hexagon, 6 squares and 6 triangles
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Dodecagon made from one hexagon, six squares and six
triangles
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Tilings from dodecagonal,
triangular and square tiles
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Tilings from dodecagons, squares and triangles
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How to prepare paper for
mosaic motives?
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How prepare the paper for tiling from hexagons, squares
and triangles
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Multi-tiles
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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
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How to pack the plane with multi-tiles?
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How to pack the plane with that kind of tiles?
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How to pack the plane with that kind of tiles
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Fractal structures
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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 …
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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 …
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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!
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Bibliography
• http://www.uwgb.edu/dutchs/symmetry/uniftil.htm
• http://en.wikipedia.org/wiki/Tiling_by_regular_polygons
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