Symmetry, Transformation and Tessellation

GeoGebra Institute of Hong Kong, Arthur Lee, Mar 16, 2018

These worksheets are extracted from course materials of MATS3210 Excursions in Geometry and Algebra with GeoGebra Session 2: Symmetries and Transformations Session 3: Tessellation

Table of Contents

  1. Single Transformation
    1. Translation
    2. Reflection
    3. Rotation
    4. Glide Reflection
    5. Dilation
  2. Combine Transformations
    1. Two reflections
    2. Two rotations
    3. Rotation and Translation
    4. Rotation and Reflection
  3. Patterns
    1. Kaleidoscope
    2. Kaleidoscope with 3 mirrors
    3. Line patterns from sequence of dilation
    4. Dilation Animated
    5. Dilations of a star
    6. Repeated dilation and rotation
    7. Repeated rotation
    8. Spiral from polygons
    9. square pattern
    10. Equiangular Spiral
    11. Equiangular Spiral on Nautilus Shell
    12. Rotating and Dilating Squares
    13. Rotating and Dilating Hexagons
    14. combining spirals
    15. Translate Dots
  4. Transformation Commands
    1. Rotation and Dilation
    2. Using sequence command
    3. More about sequence
    4. Modify triangle spiral
    5. Other spirals
    6. Different sequences
  5. Tessellation Commands
    1. Translation in 2 directions
    2. Simple tessellation
    3. Setup for tessellation
    4. Common tessellations
    5. making pentagonal tiles
  6. 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
  7. Special Tiles
    1. Rotating Squares Tiling
    2. Rotating Triangles Tiling
    3. pentagon tiling case 1
    4. pentagon tiling case 2
    5. hexagon tiling

1.

Single Transformation

  • 1. Translation
  • 2. Reflection
  • 3. Rotation
  • 4. Glide Reflection
  • 5. Dilation

1.1.

Translation

Change the original object or the vectors to explore the effect of translation.
Translation
– GeoGebra Institute of Hong Kong

1.2.

–

1.3.

–

1.4.

–

1.5.

–

2.

Combine Transformations

  • 1. Two reflections
  • 2. Two rotations
  • 3. Rotation and Translation
  • 4. Rotation and Reflection

2.1.

Two reflections

Change the original object or the mirror lines to explore the effect of two reflections
Two reflections
– GeoGebra Institute of Hong Kong

2.2.

–

2.3.

–

2.4.

–

3.

Patterns

  • 1. Kaleidoscope
  • 2. Kaleidoscope with 3 mirrors
  • 3. Line patterns from sequence of dilation
  • 4. Dilation Animated
  • 5. Dilations of a star
  • 6. Repeated dilation and rotation
  • 7. Repeated rotation
  • 8. Spiral from polygons
  • 9. square pattern
  • 10. Equiangular Spiral
  • 11. Equiangular Spiral on Nautilus Shell
  • 12. Rotating and Dilating Squares
  • 13. Rotating and Dilating Hexagons
  • 14. combining spirals
  • 15. Translate Dots

3.1.

Kaleidoscope

Change the original red dot or angle between mirrors to explore successive reflection in 2 mirrors.
Kaleidoscope
– GeoGebra Institute of Hong Kong

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

3.8.

–

3.9.

–

3.10.

–

3.11.

–

3.12.

–

3.13.

–

3.14.

–

3.15.

–

4.

Transformation Commands

  • 1. Rotation and Dilation
  • 2. Using sequence command
  • 3. More about sequence
  • 4. Modify triangle spiral
  • 5. Other spirals
  • 6. Different sequences

4.1.

Rotation and Dilation

combine rotation and dilation
In this example, a triangle (named 'poly1') is rotated and dilated with a specified angle t and factor r, about the center A. The two transformations can be combined as one in the command.[br][br]Dilate[Rotate[poly1, t, A], r, A]
repeat the action
We can continue with this action on each object formed, resulting in a series of triangles, which can be named as poly2, poly3, poly4, etc by repeating the command in the following way:[br][br]poly2=Dilate[Rotate[poly1, t, A], r, A][br]poly3=Dilate[Rotate[poly2, t, A], r, A][br]poly4=Dilate[Rotate[poly3, t, A], r, A][br]poly5=Dilate[Rotate[poly4, t, A], r, A][br]…...
figure 2
varying the parameters
Here is another way for creating each of these triangles by varying the angle and factor each time, but applying the transformation to the first figure only.[br][br]Dilate[Rotate[poly1, n*t, A], r^n, A][br][br]We use the letter n to indicate suitable angle and factor if the action is repeated on the first triangle. Drag the slider for n to see the effect.
figure 3
– GeoGebra Institute of Hong Kong

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

5.

Tessellation Commands

  • 1. Translation in 2 directions
  • 2. Simple tessellation
  • 3. Setup for tessellation
  • 4. Common tessellations
  • 5. making pentagonal tiles

5.1.

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

5.2.

–

5.3.

–

5.4.

–

5.5.

–

6.

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

6.1.

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

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

6.11.

–

6.12.

–

6.13.

–

6.14.

–

6.15.

–

6.16.

–

6.17.

–

6.18.

–

7.

Special Tiles

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

7.1.

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

7.2.

–

7.3.

–

7.4.

–

7.5.

–

Information