Non-extensible polycycles

Now we consider another natural notion of maximality for polycycles. An (r, q)-polycycle is called rum-extensible if it is not a partial subgraph of any other (r, y)-polycycle, i.e. if an addition of any new r-gon removes it from the class of (r, ) polycycles. It is clear that any tiling r, q) ( r, q) — f in elliptic case) is non-extensible, while all other non-extensible polycycles are helicenes. It is also clear that any 3-connected (r, 3)-polycycle is non-extensible. 

For the parameters (r, q) = (3,5), (5,3), in Lemma 8.2.3, we will prove that the vertex-split Icosahedron is such a unique finite polycycle.  

Proof The proof is based on curvature estimates. In fact, it is the counting of vertex-face incidences plus using the Euler formula. We choose to use the curvature viewpoint (see Section 4.4) since it express nicely ellipticity. 

Since the angle of a regular r-gon is equal to r- n and the number of regular r-gons that meet at an internal vertex of the polycycle P is equal to q, the curvature of any internal vertex of the polycycle P is equal to  

If Vi , = i.e. an (r, 7)-polycycle P is outerplanar, then the curvature 2 is equal to zero for any parameters (r, q). If vint 0, then the curvature 2 is positive, zero, or negative, depending on whether the parameters (r, q) are elliptic, parabolic, or hyperbolic, respectively. 

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We will use this curvature only for non-extensible polycycles in Section 8.2, and it will be only a counting, i.e. combinatorial, argument However, the above curvature, concentrated on points, can be made geometric. This is the subject of Alexandrov theory (see, for example, [Ale50]). 

Four exceptional non-extensible polycycles, depicted on Figure 8.3 are vertex-split Octahedron, vertex-split Icosahedron, and two infinite ones Pi x Pi = Prism<, and Tr% = APrismoo (see Section 4.2). 

The positivity of the right-hand side of Inequality (8.4) implies the positivity of the left-hand side. Thus, in view of Equation (8.1), the curvature 2 of the geodesic -gon is positive. The resulting inequality 2(r + q) — qr > 0 contradicts the assumption made. Hence, a finite non-extensible polycycle cannot have parabolic or hyperbolic parameters (r, q) these parameters are elliptic. ... 

The cross-dimerization reaction is very commonly employed for the manufacture of intermediates for synthetic musks, which have become an important class of perfumery chemicals. Synthetic musks have been the target of extensive research over the years due to a conservation order placed on the musk deer. Nitro musks are being steadily replaced by non-nitro polycyclic musks becau.se of technical drawbacks and health aspects of the former, which are explosive, sensitive, and virtually nonbiodegradable. Non-nitro musks, on the other hand exhibit better stability to light and alkali, and more nearly duplicate the odour of the macrocyclic musks occurring in nature. Indian musk odorants are easily soluble in alcohol and perfume compositions. They have the added advantage of non-discoloration in soap and domestic products. In view of the low price, their future in the perfume industry appears very promising. 

The vertex-split Octahedron and the vertex-split Icosahedron are polycycles obtained from Octahedron and Icosahedron, respectively, by splitting a vertex into two vertices and the edges, incident to it, into two parts, accordingly. The vertex-split Octahedron is drawn on Figure 4.2,1 and both of them are given on Figure 8.3 they are the only, besides five Platonic r, q) — f, non-extensible finite (r, )-polycycles. 

The determination of all non-extensible (r, )-polycycles, i.e. ones that cannot be extended by adding an r-gon. In particular, besides 5 platonic cases and 2 exceptional elliptic polycycles, they are infinite. 

Proof The case (r, q) = (3,3), (3,4), (4,3) follows immediately from the list of these polycycles given in Chapter 4. It is clear that doubly infinite and non-periodic (at least in one direction) sequences of glued copies of the elementary polycycles bi and ee, (from Figure 7.3) yield a continuum of infinite non-extensible (3,5> polycycles. By gluing the elementary (5,3)-polycycles Ci (from Figure 7.2) and C 2 (obtained from Ci by rotation through n), we obtain infinite non-extensible (5,3)-polycycles. Clearly, there is a continuum of such. In Lemma 8.2.4, we will construct a continuum of non-extensible (r, )-polycycles for non-elliptic (r, q). 

Lemma 8.23 ([DSS06]) All finite elliptic non-extensible (r, q)-polycycles are two vertex-splittings (of Octahedron and Icosahedron see first two in Figure 8.3) and five Platonic r, q) (with a face deleted). 

Consider now the case (r, q) = (5,3). It is easy to see that the only finite elementary (5,3)-polycycle, which is non-extensible, is A = 5, 3 — /. Assume now that Maj(P) has a vertex of degree 1. Then, the elementary polycycle corresponding to this vertex is different from A,. It is easy to see that for all other finite elementary (5,3)-polycycles, we can extend, i.e. add one more face. 

Consider now the case (r, q) = (3,5) and take a non-extensible (3,5)-polycycle P. The only finite elementary non-extensible (3,5)-polycycle is oi = 3,5 — f. Assume now that P is different from a, then it has more than one elementary component So, the major skeleton Maj(P) has vertices of degree 1. 

Since the polycycle P is non-extensible, there are some polycycles incident to the edges ei and ez. So, we have two paths starting from d. Since Maj(P) is a finite tree, those two paths will eventually terminate on a vertex of degree 1, which, by the above analysis, has to be another d. Furthermore, the elementary (3,5)-polycycle preceding it has to be c3. So, again we have two paths, one of them new. This argument does not terminate. We do not find a cycle in Maj(P) since it is a tree and so we proved that P is infinite. This is impossible by the hypothesis and so the only possibility is d + a3. ... 

An (R, 3)-polycycle is called totally elementary if it is elementary and if, after removing any face adjacent to a hole, we obtain a non-elementary (R, 3)-polycycle. So, an elementary (R, 3)-polycycle is totally elementary if and only if it is not the result of an extension of some elementary (R, 3)-polycycle. See below for an illustration of this notion ... 

The MOs of bi- or polycyclic aromatics like naphthalene or phenan-threne do not exhibit the orbital degeneracies characteristic of benzene, even at the primitive HMO level to cite Craig [34, p. 13] the [Hiickel] rule has no more than vestigial force [in them] . Nor has its extension to non-planar transition states, plausible as it may appear, been proven to be generally valid. 

At nearly the same time MacMillan and coworkers developed a new protocol for SOMO-catalyzed intramolecular arylation of enolizable aldehydes (Scheme 4.7) [33]. In these studies the required oxidation step was accomplished by tris-phenanthroline complexes of iron]111) bearing non-nucleophilic counterions, such as Fe(phen)3-(PFd3. Higher degrees of enanhoselectivities were obtained compared to oxidations with CAN (see 34, Scheme 4.7). Using this method a simple three-step-access to (-)-tashiromine was elaborated by the authors. For theorehcal calculahons of this transformation see Reference [34], By extension of this concept to suitable requisite tethered polyenes the authors were able to estab-hsh a powerfijl cascade reaction leading to defined configured polycyclic structures (36, Scheme 4.7). The oxidation step in this process was achieved by slow addition of Cu(OTf)2/TFA sodium salt [35]. 

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