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As seen above, I don't follow how you figure out those planes. It seems they're not using the origin labeled. I'm not really sure I understand spatially what's going on in the left figure so let's focus only on the right figure. Why is one of the intercepts not really touching the xyz axes? Where is the origin in this thing?

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You may be aware of this already but I believe these are Miller Indices - are you familiar with those? –  Mark Rovetta May 1 '13 at 21:10
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2 Answers 2

The origin is at the point where the $x$, $y$, and $z$-axes meet, as usual.

The numbers in parentheses are Miller Indices (as Mark pointed out in his answer). In the case of a simple cubic crystal, the indices $(h k l)$ represent a plane that passes through the points $(1/h,0,0)$, $(0,1/k,0)$ and $(0,0,1/l)$. More generally, the indices are the coefficients of a vector orthogonal to the plane, in the basis of primitive reciprocal lattice vectors (you can check that these two definitions are equivalent for a cubic crystal).

In the example figure on the right, the Miller indices are $(1 \bar{1} 1)$, so the plane should intersect the $x$, $y$, and $z$-axes at $1$, $-1$, and $1$ respectively. The $x$- and $z$-intercepts are explicitly shown in the figure, but the $y$-intercept is not; it would be visible if the figure was extended to the left by one unit.

Similarly, for the example figure on the left, the plane will intersect the $x$, $y$, and $z$-axes at $-1$, $1$, and $1$ respectively. Here the $y$- and $z$-intercepts are shown; the $x$-intercept would be visible if the figure was extended back into the screen by one unit. Notice the symmetry between the two cases.

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The notation in the figures looks correct to me for the planes shown if you interpret the numbers in parentheses as representing Miller Indices. Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices.

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