In books on mineralogy or crystal formation, you will often see three to four Arabic numerals in parentheses noted above each plane of a crystal. The symbol is used to indicate the spatial relationship between a mineral crystal's crystal planes so it's known as the "Crystal Plane Symbol". The crystallographic axis is used as the reference axis (imaginary line) together with the axial units above to indicate the location of the crystal plane. This is made up of 3 to 4 (cubic and hexagonal) coprime small integers in parentheses. As the notation was created by the British mineralogist Miller in 1839, it is also known as the "Miller Index".
The Miller Index for any crystal is equivalent to the inverse intercepts of that crystal plane on three or four crystallographic axes. In Image 1 for example, to find out the symbol for crystal plane ABC you must first calculate the inverse intercepts for plane ABC along the X, Y and Z axes. The result is:
1/2 ： 1/3 ： 1/6 ＝ 3 ： 2 ： 1 (The three smallest coprime integers)
The symbol for plane ABC is therefore (321).
In the standard coordinate system, the front of the X-axis is positive and behind is negative; the right side of the Y-axis is positive and the left is negative; in the Z-axis, up is positive and down is negative. If a crystal plane intersects with the negative end of a crystallographic axis, a bar must be marked above the corresponding index, so in Image 1 plane ACD's symbol is (321).
In the Crystal System the hexagonal system is unusual in that its Miller Index is made up of 4 smallest coprime integers. As a result, Beryl's Miller Index is as shown in Image 2.
Using the Isometric System as an example, the crystal in Image 3 is made up of cubic and octahedral lattices. Can you find the correct symbol for each crystal plane?