OTHER TYPES OF TESSELLATION
Section: Geometry
 
1.TesSELLATING THE PLANE WITH NON-EQUILATERAL TRIANGLES.
Any triangle will tessellate the plane. 
1.- Cut out ten or fifteen identical triangles that are not equilateral from a piece of card. Try and make them tessellate the plane. Will they tessellate? 

2.-Use what you have discovered to give a step by step explanation of how to tessellate the plane with any kind of triangle.

Prove that this is true by using points B and C to change the shape and size of the triangle. Don't forget that point A is fixed.
2. TesSELLATING THE PLANE WITH QUADRILATERALS.
Any rectangle will tessellate the plane. Any quadrilateral will also tessellate the plane. Look at the beautiful patterns that have be made using different shapes in the second window.
 

1.- You have proved that any triangle can be used to tessellate the plane. Is this also the case with any rectangle?

 

Prove that this is the case by using point B in the following window to change the shape and size of the rectangle. Remember that point A is fixed.

2.- What about using any quadrilateral? 

3.- Cut out ten or fifteen identical quadrilaterals from a piece of card and try fitting them together. Do they tessellate? 

 

Use points B, C and D to change the shapes and their sizes. Don't forget that point A is fixed.
3. TesSELLATING THE PLANE WITH PENTAGONS.

A regular pentagon will not tessellate the plane but there is an equal-sided pentagon that will.

In the following window you can see how an equal-sided pentagon tessellates the plane. This kind of tessellation is called 'Cairo tessellation' as many of the streets in Cairo were paved in this way.

1.-You have now seen how any three or four-sided polygon will tessellate the plane. You also know that regular pentagons do not tessellate; if you try and fit three regular pentagons together there is a gap where the vertices meet and the shapes overlap if you try to fit four together. Is there a five-sided polygon that will tessellate? 

We call mosaics formed using irregular polygons simple tessellations of non-regular polygons.
Change the size by moving point A  

2.- Try fitting together different equal-sided polygons until you find one that tessellates. If you can't then make the sides different lengths (e.g. like the shape of a "house").
4. TesSELLATING THE PLANE WITH DISTORTED REGULAR MOSAICS (MOVING AWAY FROM REGULARITY).

Another type of tessellation is made by distorting a regular mosaic by changing one of the sides of the regular shape. The distortion just needs to keep a certain degree of symmetry.
 

1.-Move points E and F in the first window to change the shape and the tessellation pattern produced. Try not to let any of the lines cross over each other.
Points A, B, C and D are fixed. 

2.-We are now going to do the same thing using a regular mosaic made of equilateral triangles as our starting point. In the second window move points D, E and F to change the shape and the tessellation pattern produced. Try not to let any of the lines cross over each other.

3.-Find out as much as you can about the life and work of the brilliant artist Maurit Escher. Symmetry is beauty.

 
 
         
           
 

Ángel Aguirre Pérez - aap@sauron.quimica.uniovi.es
 
Spanish Ministry of Education. Year 2001
 
 

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