Tessellations Java Version

A tessellation is an arrangement of tiles which fit together exactly and completely cover a flat surface.
Here are two tessellations:

4 squares per vertex = 44

Triangle hexagon triangle hexagon at each vertex. = 3.6.3.6

The hexagons and the equilateral triangles have sides that are the same size. Do they look the same? The hexagons' sides look bigger but aren't really.

Task 1 Tiling the plane with regular polygons.

There are three polygons that can be used on their own to tile the plane completely.
Squares are easiest to use so try them first.
Have you worked out what the other two shapes are?
Can you get them to join together and produce a tiling?

Left click and drag to move a polygon.

Right click and drag to rotate a polygon.

Task 2. Tiling the Plane: part 2.

You can also tile the plane by using combinations of regular polygons. There are 8 ways that you can do this. Each vertex must have the same number of the same type of polygon and the polygons must all have the same size side. Here is one of the patterns.
To find the other 7 patterns first find sets of polygons that can be joined together around one vertex without overlapping or leaving gaps. This example fits 2 squares, a triangle and a hexagon around one vertex. The problem is that you can't join this pattern up without having different arrangements at some of the vertices.

Useful Data

Here are the internal and external angles of the regular polygons that are shown in the applet.
None of the other regular polygons produce complete tilings and, in fact, only 5 of these will work!.

Remember that the angles at a vertex must add up to 360 degrees!

Task 2. Tiling the Plane: part 3.

If you alter the rules to allow vertices to have different arrangements of polygons then you can produce a huge number of tilings.

Investigate tilings that completely cover the plane but feature two or more tile arrangements at their vertices.

Is it possible to tile the plane with regular polygons in such a way that the pattern never exactly repeats?

This pattern has three vertex arrangements. Consider the vertices of the triangles and count clockwise. The arrangements are: 3.6.42, 3.42.6. and 3.6.3.6
This pattern has several vertex arrangements. If you study it closely you might see how it relates to other patterns that you have already made. 3.6.42, 3.42.6. and 3.6.3.6

To produce HUGE patterns

This webpage uses a Java applet. You can use Logo to play with these patterns. You can also assemble parts of a pattern using this applet and then put them into paint to build a huge tiling.
To export a picture to paint press Prt Scrn and then open Paint and paste into it.