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:
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.