Angles and curvature

Recall that the regular tiling [r, q) lives on X = 2, M2, or H2 according to whether parameters (r, q) are elliptic, parabolic, or hyperbolic. The r-gons are regular in X and their curvature is 2ra(2, r, q). 

If x and y are two points in p(P), then there exists an integer no such that x, y e p(Dec(P) ). The geodesic d between x and y is included in 0(Dec(f ) )) and, therefore, just as well in p(P). If P is not proper, then there exist two distinct vertices v, v (or edges, faces) of P, whose images in r, q) coincide. There exists an integer o such that v, v Dec(P) 0. But Dec P)nt is proper, so their images in r, q do not coincide.  

In the proof of the above theorem, we need to use finite polycycles because infinite polycycles can have an infinity of boundaries. 

Proot By the proof of Theorem 4.4.1, we can assume, Without loss of generality, that P is finite. Denote by 0 the cell-homomorphism in 3, q+2) and assume further that P is not a proper (3, q + 2)-polycycle. Then we can find two vertices v, v on the boundary of P with p(v) = p(i/) and the image of the boundary path V = 

that for ft = 7, there are outerplanar (3,4)- and (3,5)-polycycles, which remain helicenes in 3, 5 and 3,6, respectively. A fan of (q - 1) r-gons with -valent common (boundary) vertex, is an example of outerplanar (r, )-polycycle, which is a proper non-convex (r, 2q — 3)-polycycle. 

---

Fig. 4.35. Correlation between pyramidalization angle and curvature of the carbon cluster. Reproduced with permision from Haddon RC (1992) Acc. Chem. Res. 25 127...

---