MATS3210-2 Symmetries and Transformations

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

MATS3210 Excursions in Geometry and Algebra with GeoGebra Session 2: Symmetries and Transformations [color=#c51414]Note for this Collection:[/color] https://dl.dropboxusercontent.com/u/4210789/me/mini/tiling/index.html

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

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.

–

Information