= Solution
Let $\mathbf a_1,\mathbf a_2$ be equal-length primitive translation vectors of the <graphene> lattice, separated by $60^\circ$. The <carbon nanotube chiral vector>
$$
\mathbf C_h=n\mathbf a_1+m\mathbf a_2
$$
identifies the two points joined when the sheet is rolled: $|\mathbf C_h|$ is the circumference, not the tube axis. The right-hand panel of the preceding original diagram shows $\mathbf C_h=6\mathbf a_1+6\mathbf a_2$ and $\mathbf C_h=7\mathbf a_1$.
The periodic boundary condition is $\mathbf k\cdot\mathbf C_h=2\pi\ell$, with $\ell$ an integer. Applying it at either <Dirac valley> gives the <nanotube zone folding> criterion
$$
n-m\equiv0\pmod3
$$
for a metallic tube. Thus $(6,6)$ is an armchair metallic example, whereas $(7,0)$ is a zigzag semiconducting example. Other choices with $n-m\not\equiv0\pmod3$ also give a <semiconducting carbon nanotube>.
This criterion is the ideal flat-sheet approximation. Curvature can open a small gap in nominally metallic non-armchair tubes; ideal armchair tubes remain gapless in that approximation. The property is determined by the wrapping vector, not simply by calling a tube “chiral” or by its diameter alone.
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