session 2 supplement
This book provides examples and tips about commands for combining and repeating transformation in the creation of patterns.
Table of Contents
- spiral patterns
- Rotation and Dilation
- Using sequence command
- More about sequence
- Modify triangle spiral
- Other spirals
- repeated translations
- Translation in 2 directions
- Simple tessellation
- Setup for tessellation
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
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]