Space filling

We can think of building up a crystal from a set of identical building blocks or polyhedra. The shape of these blocks is fixed by the atoms from which they are made. Some are cubic like this one, with an atom at each corner of the cube.

Cubes have square sides. If some of the sides are rectangles they can still pack together to fill space. Do you think that all space filling blocks have sides which meet at right angles? Some insects build space filling blocks in their nests; for example bees store honey in hexagonal honeycombs. They look like the left hand diagram below:
The right hand diagram shows a view of a cube rotated until it has a hexagonal outline like the honeycomb shapes. Do you think that honeycombs are made from cubes but we do not realise this because we do not look at them from the correct direction.? Here is a set of cubes filling space which look similar to the honeycomb. There may be other shapes which can fill space. If you were a bee you would want to enclose the biggest volume you could make from the smallest area of beeswax. What about this shape? The sides have 4 corners, but unlike a cube which always has just 3 sides meeting at each corner, this shape has some corners with 4 sides meeting. The diagram below shows the 3 sides meeting at a corner on the left, the middle diagram shows it after a small rotation, and on the right the rotation continues until you can only see the 4 sides meeting at a corner, when the outline has become square. So this shape sometimes looks similar to a cube which has a square outline when seen from some directions. This shape is called a rhombic dodecahedron.

The name comes from the ancient Greeks; dodec meant 12, hedron meant a solid shape, so dodecahedron is a solid with 12 faces, rhombic describes the shape of each face, they have 4 sides but they are not squares.

More spacefilling polyhedra can be seen on another page

Click here for instructions on how to make your own models and try fitting them together to see how well they fill space.