Structure matters

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Packing atoms

     Solid matter is built up from the packing together of atoms which have the shape

of spheres.  There is a direct correlation between the way that spheres pack together

and how polyhedra pack together to fill space. 

     For example, a layer of equal radius spheres (atoms) can be packed together to

fill the plane in two characteristic ways. Observe that the void spaces, or interstices,

between the spheres in the triangular pattern are smaller than those in the square

pattern.  This is because the spheres in the triangular pattern nestle into the

depressions formed between them.  As a result the spheres in the triangular array

are packed together closer than those in the square packing.  Indeed this is the

closest packed arrangement of spheres possible in two dimensions.  If the center

points of the spheres, representing the nuclei of atoms, are connected together by

lines, two dimensional grids, or lattices, are formed with the sphere centers (nuclei)

represented by the vertices of the lattice.  Lattices, in turn, can be modeled as nets,

or tessellations, of polygons which are linked together to fill the plane as shown in

Figure 31.

     Notice that the pattern of points surrounding every lattice point is identical, or

isometric.  Such lattices have translational symmetry.  That is, a copy of one section

of the lattice can be superimposed on another section so that their points coincide

completely without changing the pattern.  Hence they have utility in modeling the

structure of crystals since most solid matter is almost always spatially periodic.

     Representing the arrangement of atoms in a substance by the vertices, edges,

and faces of polygons is called the topological model of matter because topology is

concerned with the symmetry of these geometric elements.  It enables the geometry

and symmetry of atomic structures to be visualized more clearly.  Therefore the

topological model of matter is the convention used throughout this lesson

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