MATS3210-3 Tessellations

GeoGebra Institute of Hong Kong, Arthur Lee, Nov 24, 2014

MATS3210 Excursions in Geometry and Algebra with GeoGebra Session 3: Tessellations [color=#c51414]Note for this collection:[/color] https://dl.dropboxusercontent.com/u/4210789/me/mini/tiling/index.html

Table of Contents

  1. commands illustrated
    1. Translation in 2 directions
    2. Simple tessellation
    3. Setup for tessellation
    4. Common tessellations
  2. wallpaper symmetry
    1. Download Wallpaper Files
    2. p1
    3. p2
    4. p3
    5. p3m1 (kaleidoscope 60)
    6. p31m
    7. p4 (pinwheel)
    8. p4m
    9. p4g
    10. p6
    11. p6m (kaleidoscope 30)
    12. pm
    13. pg
    14. pmm
    15. pgg
    16. pmg
    17. cm
    18. cmm
  3. special tiles
    1. Rotating Squares Tiling
    2. Rotating Triangles Tiling
    3. pentagon tiling case 1
    4. pentagon tiling case 2
    5. hexagon tiling

1.

commands illustrated

  • 1. Translation in 2 directions
  • 2. Simple tessellation
  • 3. Setup for tessellation
  • 4. Common tessellations

1.1.

Translation in 2 directions

translation by combination of vectors
An object can be translated with various combinations of given vectors. For example, the following shape (poly1) is translated by the vector u+2v. The command is:[br][br]Translate[poly1, u+2v][br][br]This means that poly1 is moved by the vector u, and then further moved by the vector 2v. Change vector u or v to see how the movement is affected.
figure 13
translate repeatedly in the same direction
We create a list of objects by translating ploy1 using different multiples of vector u (from -3u to 3u). [br][br]The command is:[br][br]list1=Sequence[Translate[poly1, m*u], m, -3, 3][br][br]The result is a set of 7 copies called list1. This list1 can then be transformed in another way as a single object.
figure 14
moving in another direction
The list1 created in previous example can be moved in another direction by the vector v. The slider k is used to illustrate how this result may change by using different multiples of vector v.[br][br]The new list is created by the command:[br][br]Translate[list1, k*v]
figure 15
array of objects
An array of objects can be made by generating another list, which contains multiple translations of previous object 'list1'. The command is:[br][br]Sequence[Translate[list1, n*v], n, -3, 3]
figure 16
– GeoGebra Institute of Hong Kong

1.2.

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1.3.

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1.4.

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2.

wallpaper symmetry

  • 1. Download Wallpaper Files
  • 2. p1
  • 3. p2
  • 4. p3
  • 5. p3m1 (kaleidoscope 60)
  • 6. p31m
  • 7. p4 (pinwheel)
  • 8. p4m
  • 9. p4g
  • 10. p6
  • 11. p6m (kaleidoscope 30)
  • 12. pm
  • 13. pg
  • 14. pmm
  • 15. pgg
  • 16. pmg
  • 17. cm
  • 18. cmm

2.1.

Download Wallpaper Files

This section contains examples for 17 wallpaper symmetry patterns. [br][br]Some applets may run very slowly. The GeoGebra files can be downloaded with this link:[br][br][url]https://www.dropbox.com/sh/ym2b8h3a1s75uxy/AAAxSZfYmOCwAES0ZP_RCXyla[/url][br][br]More information about these patterns can be found on the web, such as [br][br][url]http://en.wikipedia.org/wiki/Wallpaper_group[/url]
– GeoGebra Institute of Hong Kong

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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2.11.

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2.12.

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2.13.

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2.14.

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2.15.

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2.16.

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2.17.

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2.18.

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3.

special tiles

  • 1. Rotating Squares Tiling
  • 2. Rotating Triangles Tiling
  • 3. pentagon tiling case 1
  • 4. pentagon tiling case 2
  • 5. hexagon tiling

3.1.

Rotating Squares Tiling

This group of squares around a parallelogram can be extended to fill the plane.
A pentagon tiling can be generated from this pattern of squares.
– GeoGebra Institute of Hong Kong

3.2.

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3.3.

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3.4.

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3.5.

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Information