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MATS3210-2 Symmetries and Transformations

GeoGebra Institute of Hong Kong, Aug 1, 2014

MATS3210 Excursions in Geometry and Algebra with GeoGebra Session 2: Symmetries and Transformations

Single Transformation
1. session handout
2. Translation
3. Reflection
4. Rotation
5. Glide Reflection
6. Dilation
Combine Transformations
1. Two reflections
2. Two rotations
3. Rotation and Translation
4. Rotation and Reflection
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
Transformation Commands
1. Rotation and Dilation
2. Using sequence command
3. More about sequence
4. Modify triangle spiral
5. Other spirals
6. Different sequences

Information

Single Transformation

session handout

Combine Transformations

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

Two reflections

Change the original object or the mirror lines to explore the effect of two reflections

Two reflections

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

Kaleidoscope

Change the original red dot or angle between mirrors to explore successive reflection in 2 mirrors.

Kaleidoscope

Transformation Commands

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. 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: poly2=Dilate[Rotate[poly1, t, A], r, A] poly3=Dilate[Rotate[poly2, t, A], r, A] poly4=Dilate[Rotate[poly3, t, A], r, A] poly5=Dilate[Rotate[poly4, t, A], r, A] …...

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. Dilate[Rotate[poly1, n*t, A], r^n, A] 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.

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MATS3210-2 Symmetries and Transformations

Table of Contents

Information

Single Transformation

session handout

Combine Transformations

Two reflections

Two reflections

Patterns

Kaleidoscope

Kaleidoscope

Transformation Commands

Rotation and Dilation

combine rotation and dilation

repeat the action

figure 2

varying the parameters

figure 3