This page gives some suggestions for setup to produce tessellation by translating a figure in 2 directions. We start with any figure, which may be several objects grouped together (let us call it a cell). With 2 vectors in different directions, this cell can be translated repeatedly to fill up the plane. In later examples, a cell can be transformed to make a bigger unit (which we may call a tile) before translating. This will then results in pattern with more interesting symmetries.
In this first simple example, four line segments (a to d) are grouped together as a single object before translating. We define a list of objects with the GeoGebra command below (with the name 'cell'):
cell={a,b,c,d}
This should be entered in the input bar. You can see a new object called cell is added in the algebra window, but not shown in the geometry window. This new object can be transformed in any ways we do with other simple objects. In this example, it is translated repeatedly by the two vectors u and v.
Sequence[Sequence[Translate[cell, m*u+n*v], m, -3, 3], n, -3, 3]
In the figure, this new list of objects from translation is called 'tiling'. Drag to change the original segments and see what happens to the pattern.

figure 20

Combining cells

In figure 21, the vectors u and v define a parallelogram frame. The original cell is renamed as cell1. It is rotated by a half turn about the midpoint (M) of the parallelogram to produce another object cell2. A bigger unit (called tile) is formed by grouping cell1 and cell2 before translation.
cell2=rotate[cell1, 90, M]
tile={cell1, cell2}
This new object, tile, becomes the unit to be translated in the same way shown in figure 20.
Sequence[Sequence[Translate[tile, m*u+n*v], m, -3, 3], n, -3, 3]

figure 21

square frame and 4 fold rotation

In figure 22, the pattern is further modified by changing the frame (defined by u and v) to a square and rotate cell1 about the midpoint M by 4 right turns before translating. A new tile is made by grouping 4 cells (created from the rotation of cell1).

figure 22

square frame and reflection symmetry

In figure 23, the cells are produced by reflection about the line through M, perpendicular to the vectors. Check how the symmetries of the entire pattern are changed.