= Solution
The real-space carbon honeycomb has two atoms in a primitive cell of its triangular <Bravais lattice>. The <reciprocal lattice> is also triangular, and its Wigner-Seitz cell is a hexagonal first <Brillouin zone>. With a consistent orientation convention, \b[the reciprocal-zone hexagon is rotated by $30^\circ$ relative to a real-space carbon-ring hexagon]; reciprocal lengths scale inversely with the direct lattice spacing.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2-nanotube-reciprocal.png]
{title=Real carbon rings, the rotated graphene reciprocal zone and circumferential momentum quantization}
{source=paper-2-nanotube-reciprocal.py}
Rolling into a tube imposes $\mathbf k\cdot\mathbf C_h=2\pi\ell$, leaving parallel allowed lines in <graphene>'s two-dimensional reciprocal space. They run along the tube-axis direction, perpendicular to the circumferential wrapping vector. The nanotube's actual one-dimensional <Brillouin zone> is an interval of axial <wave vectors> with period $2\pi/|\mathbf T|$, where $\mathbf T$ is its axial translation. The parent hexagon is therefore a zone-folding construction, not a literal two-dimensional <Brillouin zone> for the finished one-dimensional tube.
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