Tessellating with Quadrilaterals.

A tessellation is an arrangement of tiles fitting exactly together which can be extended as far as required in any direction.
Here are two tessellations:

Tiling with Squares and Rectangles.

First make a square. Paint has a tool for making rectangles but it doesn't have one for drawing exact squares. Here is one way.

Open paint and select Image and then Attributes.
Deselect Draw Opaque.

In the Attributes box set the image size to 150 by 150 pixels.

Now colour this square in but keep it reasonably symmetrical.

Make sure you save it as somewhere safe with a sensible name.

Now select and copy the whole of your square and paste several copies of it onto a new Paint document.

You can tesselate the plane by using exact copies of your square or by building rectangles from it using symmetry transformations such as reflection or rotation.

Translation

Translation and Reflection.

Translation and Rotation.



Tiling with Other Quadrilaterals.

Any quadrilateral will tile the plane. You have done this with squares. Try the following shapes. These are arranged in order of difficulty.

  • Rectangle
  • Rhombus
  • Parallelogram
  • Trapezium
  • Kite
  • Dart
  • Irregular Quadrilateral

  • This picture shows the start of a tiling that uses darts. If you examine it then you can see how to tile using rhombi and parallelograms.

    Tiling with Irregular Quadrilaterals

    Any quadrilateral will tile the plane.
    Make a basic quadrilateral and make two rotated copies of it and one that has been translated.
    Then arrange them as shown.


    Join the quads together.


    Make lots of copies and join them together.


    Now colour the vertices with different colours and the body in yet another colour.
    What do you notice about the colours of the vertices on individual quads.
    What does this tell you about why this particular tiling works?

    Now develop a tile which produces an interesting pattern when you tesselate it. You might want to start with a photograph and cut sections out of it.