Relating carbon tubules

Two methods for relating different carbon tubules are described. A single graphene ribbon of infinite length but finite width can be coiled to form an infinite set of tubules, each of which has a unique pitch or helicity. This maps in a one-to-one fashion the translation/rotation operations of the group of each tubule in the set. Within the set all irreducible representations are collected into the same number of bands. Alternatively, tubules can be imagined to be pressed flat so that centers of six-member rings lie along the crease. The direction of their creases on a graphene sheet relate tubules having the same helicity but different numbers of identical rotationally symmetric subunits around their circumference. These sets help to reconcile the different expressions for band structure of tubules. These sets also help sort the various ways to join semi-infinite tubules. A perfect infinite tubule is composed entirely of hexagons. Adding one heptagon and one pentagon transforms half of the tubule into a different tubule. The two ways to group tubules suggest that the least distortion of neighboring hexagons occurs if the heptagon and pentagon are joined together or are separated to opposite sides of the tubule. In the latter case, the tubule could be imagined to be flattened so that the heptagon and pentagon are folded in half, one along each crease. This heptagon-pentagon defect best connects sets of tubules in a pairwise fashion. The paired sets of tubules have axis vectors that meet at a 30^ angle on a graphene sheet. This analysis and experimental considerations suggest that the ideal bend in a tubule caused by a heptagon-pentagon pair is likely to be 30^. Because entire sets of tubules are joined in similar fashion, tailoring of tubule electronic properties can be imagined.