In this activity, we will reinterpret the artwork [i]Concreção 8215[/i] using the [b]Sequence[/b] command in [b]GeoGebraScript.[/b][br][br][b]First Steps[/b][br][list=1][*]First, we will create a square q1 with vertices at the points A(0,0), B(1,0), C(1,1), and D(0,1). To keep the construction clear and uncluttered, remove the labels from the segments and the polygon if they are visible.[/*][*]It is recommended to open a second viewing window so that the geometric construction of the reinterpretation does not overlap with the final result.[/*][/list]

Observe that [i]Concreção 8215 [/i]is a composition formed by rotated and translated squares, arranged in rows and columns, resulting in a larger square composed of multiple smaller units.

To create a sequence of points using GeoGebraScript, we will use the [b]Sequence [/b]command in the following form:[br][br]Sequence(Expression, Variable, Initial Value, Final Value).[br][br]With this command, we can generate either a row of squares or a grid containing multiple squares arranged in rows and columns, as seen in this artwork. This allows for an efficient replication of the geometric pattern present in the composition.[br][br][list=1][*]We will create a set of points arranged in 13 rows by 13 columns. To do this, we define n = 13 as the number of points to be created per row and column.[/*][*]After entering n = 13 in the Algebra menu, a natural number slider is automatically created, highlighting n = 13. This allows the value of n to be adjusted according to the configuration set for the slider.[/*][*]To create the 13[sup]2[/sup] points, we need to specify their coordinates, providing both their x-coordinate (abscissa) and y-coordinate (ordinate). To achieve this, we use the Sequence command:[/*][/list][br][code]L = Sequence((Mod(i,n),Division(i,n)),i,0,n²-1)[/code]

After the sequence of points is created, you should be able to visualize the ordered pairs as shown in the following image.

We will use the created points as the centers of squares that result from translations and rotations of the square [i]q1[/i], considering [i]E[/i] as the center of this square.[br][br]To do this, we can use the [b]Sequence [/b]command as follows:[br][br]M = Sequence(Translate(Dilate(Rotate(q1, (x(L(i))+y(L(i)))(-15°), E), 0.5, E), L(i)), i, 1, n²)*

Next, we present an interactive activity developed using the commands discussed throughout this task. Explore it by clicking the available buttons, following the sequence of object creation.[br][br][b]Step 1:[/b] Start by generating translations, rotations, and homotheties of the original polygon.[br][br][b]Step 2:[/b] Create a reinterpretation of the artwork [i]Concreção 8215.[/i][br][br]We invite you to interact with the activity, creating your own reinterpretations, including variations in the shape of the original polygon to explore new compositions.