This is a polyhedron that fills three-dimensional space with no gaps. I discovered it in September 2008, when I started fiddling with a partial differential equation used in pattern formation. The polyhedron has 13 faces in total, of which 6 are quadrilaterals, 6 pentagons and 1 is a hexagon. As you can see from the first two pictures it has a 3-fold rotational symmetry. Looking at the edge connectivity, it seems that three edges always meet at each vertex. But that isn't true. Consider the vertex at the extreme left in the first picture, the top view. This is connected to four distinct edges. Can't you see it? The edge you're missing is shown in the bottom view (second picture). The third picture shows the arrangement of the polyhedron inside its periodic unit. Note that the centre of the hexagon coincides with the centre of the cube. Now add six more polyhedra, three of the same kind of the first one and three obtained as their mirror image. Now you can more clearly see the 4-valent vertices. The last polyhedron will sit in the remaining slot.
If you have Flash click here for further information. If you have Java click here to see a three-dimensional model of this unit.