A set of three numbers that characterize a face of a crystal. The French mineralogist René Just Haüy (1743–1822) proposed the law of rational intercepts, which states that there is always a set of axes, known as crystal axes, that allows a crystal face to be characterized in terms of intercepts of the face with these axes. The reciprocals of these intercepts are small rational numbers. When the fractions are cleared there is a set of three integers. These integers are known as the Miller indices of the crystal face after the British mineralogist William Hallowes Miller (1801–80), who in 1839 pointed out that crystal faces could be characterized by these indices.

If a plane is parallel to one of the crystal axes then its intercept is at infinity and hence its reciprocal is 0. If a face cuts a crystal axis on the negative side of the origin then the intercept, and hence its reciprocal, i.e. the Miller index for that axis, are negative. This is indicated by a bar over the Miller index. For example, the Miller indices for the eight faces of an octahedron are (III), (ĪII), (IĪI), (IIĪ), (ĪĪI), (IĪĪ), (ĪIĪ) and (ĪĪĪ).