If you divide a plane or a sphere into regions, you only need 4 colours if you wish no adjacent regions to have the same colour.

On a torus, you need 7 colours. Here is a model you can make from paper to show 7 coloured regions, which all touch each other. Drag the torus with your mouse to see all around it.

(3-D graphics courtesy of the LiveGraphics3D Java applet ).

The torus is a hexagonal ring, with a pentagonal cross-section. Click here for a pdf file which you can print to make the torus from paper. Cut out all round the coloured bits and assemble with clear tape (or leave tabs and glue them).

page date: 21Aug06.      I enjoy correspondence stimulated by this site. You can contact me here.