Steven Dutch, Natural and Applied Sciences, University of Wisconsin  Green Bay
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The rhombohedral (R) cell presents special problems in visualization. To bring out the hexagonal nature of its lattice, and the relationship between rhombohedral crystals and ordinary trigonal or hexagonal crystals, the R cell is often represented as at right, with two interior points equally spaced along a diagonal. It is the only Bravais lattice cell with more than one interior point.  
At left is a side view of a rhombohedron, which can be considered a cube stretched or flattened along one diagonal axis. A rhombohedron has six identical faces, all rhombuses. At right is a top view showing the threefold symmetry of a rhombus. Red lines denote hidden edges.  
The vertices of a rhombus fall onto equallyspaced planes as shown at left. At right is a top view.  
Here we see a layer of rhombohedra viewed from the top.  
Remember, the edges of a unit cell aren't
physically real, only the points within the cell (the
atoms) are. We may find it convenient sometimes to draw
unit cells with corners not coinciding with atoms,
especially if it brings out the symmetry better, but we
can always construct a lattice with atoms at the corners
of the unit cells. Here we remove the rhombohedron edges and construct a new set of edges defining 60120 rhombuses. We see that they enclose two equallyspaced points on a line sloping down from one corner to the diagonallyopposite corner below. Thus the two cells define the same lattice. 

Here we see how rhombohedra relate to the Rcell. Blue points are points of the Rcell, purple points are in front of it, green ones behind, and gray in the same plane but outside the Rcell. 
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Created 18 September 1998, Last Update 18 September 1998
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