Explain close packing in one dimension. Explain close packing in two dimensions.

(N/A) The spheres in one dimension can be arranged only in one way,that is,to arrange them in a row and touching each other.
Close packing of spheres in one dimension:
In this arrangement,each sphere is in contact with two of its neighbours. The number of nearest neighbours of a particle is called the coordination number. Thus,in a one-dimensional arrangement,the coordination number is $2$.
The close packing in two dimensions can be done in two ways by stacking the rows of closed-packed spheres: $(i)$ Square Close Packing,$(ii)$ Hexagonal Close Packing $(HCP)$.
$(i)$ Square Close Packing: In square close packing,the spheres of the second row are placed exactly above the spheres of the first row such that the spheres of these two rows are aligned horizontally as well as vertically.
If the first row of spheres is called $A$ type row,the second row being exactly the same as the first one is also of $A$ type and hence by placing more rows,$AAAA...$ type of arrangement is obtained.
In this arrangement,each sphere is in contact with four of its neighbours and hence the coordination number of a sphere is $4$.
If the centres of the four immediate neighbours are joined,a square is formed and hence it is named Square Close Packing.
$(ii)$ Hexagonal Close Packing $(HCP)$:
In hexagonal close packing,the spheres of the second row are fit in the depressions of the spheres of the first row.
If the spheres of the first row are called $A$ type,the ones in the second row may be called $B$ type. The spheres of the third row are placed in the depressions of the spheres of the second row such that the spheres of the first row and third row are in the same alignment horizontally and vertically.
As the third row is exactly the same as that of the first row,it is called $A$ type. Hence,we get $ABABAB...$ type of arrangement.
In this arrangement,each sphere is in contact with six of its neighbours and thus in two dimensions,the coordination number is $6$. The centres of these six spheres are at the corners of a regular hexagon,hence this packing is called two-dimensional hexagonal close packing.