Creating a Tessellations Activity

An interesting activity is to cover a page with a tessellation of a picture. This lets us explore working with images and also look at using sequences.

Open a blank GeoGebra page.

We want to start with some technical preparation.

Use the Point two to create two points, A and B.

Use the commands “a=Vector[A]” and “b=Vector[B]”.

(The technical point is that we will want points for placing and sizing the first image and vectors for defining translations.)

From the slider menu of the toolbar, select the insert image tool.

We then selected an image of the GeoGebra symbol. This got loaded as pic1. Next we want to fix the size and location of the image.

Either right-click(pc) or crtl-click (mac) on the image to open the contextual menu and select the object properties.

Use the position tab and set three of the corners of the base picture. Set Corner 1 at (0, 0), Corner 2 at A, and Corner 4 at B.

Note how with the given positions of A and B we get a stretched version of the picture.

Next we want to set up copies of the picture with the sequence command. We will make copies that translate by 2*a each time. The command is

“ImageList1=Sequence[Translate[pic1,2*i*a],i,-10,10]”.

Now we translate the image to fill the screen. We do this with a double use of the sequence command, with one direction shifting by multiples of 2*a and the other direction shifting by multiples of (a+b).

ImageList2=Sequence[Sequence[Translate[pic1,i*2*a+j*(a+b)],i,-10,10],j,-10,10]

This arrangement fills in every other square. If we wanted to fill in every square we simply translate by multiples of a and b.

ImageList3=Sequence[Sequence[Translate[pic1,i*a+j*b],i,-10,10],j,-10,10]