This wonderful thing was discovered by Szilassi in the 1970s. It has seven faces, each of which is a hexagon (that is, each face has six sides). Each face touches every other face. You therefore need 7 colours if each face is coloured differently to a face it touches. You may know that you need at most 4 colours to colour any map on a plane or sphere, so that no two adjacent areas are the same colour (Four Colour Theorem). Here we have effectively a torus (ring doughnut shape) rather than a sphere; a torus needs at most 7 colours.

In the 3-D rendering below, drag the polyhedron with your mouse to see all sides. (This uses the very fine LiveGraphics3D Java applet ).

Your best bet for more information is http://minortriad.com/szilassi.html,  which is where I discovered it. There are some nice 3D pictures, and a quality net so you can make it. It's also worth looking at MathWorld.

On the other hand, if you'll settle for lesser quality, I've made this net that is (I think) a lot easier to construct. Instructions are below it:

How to make:
I suggest using paper or card, secured together with sticky tape.
1. Print out the net (save it to file first if you wish, or click here to open it in a new browser window for direct printing)
2. Cut out each of the 4 shapes - the solid lines indicate the faces, but I recommend leaving the bit shown by the dotted lines to make it easier to build.
3. Fold along the lines of the shape A
4. Attach faces B, C, D to shape A as shown by the arrows labelled '4'. Shape A overlaps (on top of) the smaller shapes.
5. Fold the shape in, so that the two corners labelled 5 come together. Secure the solid lines that these corners lie on together. Once you crack this, the rest is plain sailing. You may also wish to secure the other faces which have been brought together by this.
6. Fold face D right up flat against shape A; fold the face indicated by arrow 6 till it touches two other faces, and secure it.
7. Fold down face A, and secure where it touches 4 other faces
8. Fold the last face into position and secure.

If you wish to see 7 colours on a more obvious torus, have a look at this 3-D model which can also be made from paper.


page date: 23Oct05.      I enjoy correspondence stimulated by this site. You can contact me here.