Proper polycycle

Consider now the notion of reciprocity, defined for some proper polycycles. Let P be a proper bounded (r, q)-polycycle. Consider the union of all r-gonal faces of r, q outside of P. Easy to see that this union will be an (r, 4)-polycycle call it then a reciprocal polycycle to P if either P is elliptic or P is infinite and has a connected boundary. Call a polycycle self-reciprocal if it admits the reciprocal polycycle and is isomorphic to it. 

Carbonyl condensation reactions are perhaps the most versatile methods available for synthesizing complex molecules. By putting a few fundamental reactions together in the proper sequence, some remarkably useful transformations can be carried out. One such example is the Robinson annulation reaction for tire synthesis of polycyclic molecules. The word annulation comes from the Latin annulus, meaning "ring," so an annulation reaction builds a new ring onto a molecule. 

Polyene Cyclization. Perhaps the most synthetically useful of the carbo-cation alkylation reactions is the cyclization of polyenes having two or more double bonds positioned in such a way that successive bond-forming steps can occur. This process, called polyene cyclization, has proven to be an effective way of making polycyclic compounds containing six-membered and, in some cases, five-membered rings. The reaction proceeds through an electrophilic attack and requires that the double bonds that participate in the cyclization be properly positioned. For example, compound 1 is converted quantitatively to 2 on treatment with formic acid. The reaction is initiated by protonation and ionization of the allylic alcohol and is terminated by nucleophilic capture of the cyclized secondary carbocation. 

The idea that such cyclisations should be highly stereoselective derives from the Stork-Eschenmoser hypothesis, according to which the double bonds in polyalkenes of the squalene type are properly arranged to undergo cyclisation to fused polycyclic systems with the natural trans-anti-trans configuration. 

By proper choice of reaction conditions (metal, solvent, the order of addition of reagent and reactant), fused polycyclic aromatics can be converted to different partially reduced derivatives with high selectivity. When the red complex of naphthalene, for example, formed in the Na-NH3 solution, is quenched with aqueous ammonium chloride solution, 1,4-dihydronaphthalene is formed209 [Eq. (11.58)]. Reaction in the presence of an alcohol yields 1,4,5,8-tetrahydronaphthalene210 [Eq. (11.59)]. The Benkeser reduction affords further saturation to the isomeric octalins211 [Eq. (11.60)] or even to decalin212 [Eq. (11.61)] ... 

Call a polycycle proper if it is apartial subgraph of r, q) and a helicene, otherwise (this term will be justified later by Theorem 4.3.1). Call a proper (r, )-polycycle induced (moreover, isometric) if this subgraph is, in addition, an induced (moreover, isometric) subgraph of r, q). Another interesting possible property of aptoper (r, q)-polycycle is being convex in r, q (see Section 4.4). 

There is a literature (see, for example, [GrSh87a, Section 9.4], [BGOR99], [BCH02], and [BCH03]) about proper parabolic polycycles (polyhexes, polyamends, polyominoes for 6,3, 3,6), 4,4, respectively) the terms come from familiar terms hexagon, diamond, domino, where the last two correspond to the case p, ... 

The number of (3,4)-polycycles is also countable, including two infinite ones (9 of (3, 4)-polycycles are proper and 5 of proper ones are induced). 

Note, 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. 

List of 4 sporadic elementary ( 2,3,4,5, 3>polycycles with 1 proper face ... 

List of 36 sporadic elementary ( 2, 3,4, 5], 3)-polycycles with 5 proper faces ... 

For (5,3)-polycycles with x < 11, Ns x) was found in [CCBBZGT93] all such extremal (5,3)-polycycles turn out to be proper and unique. Moreover, the (S, 3)-polycycles, which are reciprocal (see Section 4.1) to any such extremal ones, turn out to be also extremal. [CCBBZGT93] asked about N ix) for x > 12 this section answers this question for any x and for all elliptic (p, q). 

For parabolic or hyperbolic parameters (r, q), the spiral Sp n) is defined as the proper (r, q)-polycycle with n r-gons obtained by taking an r-gon and adding r-gons in sequence, always rotating in the same direction. A formal definition is difficult to write (see [HaHa76, GreOl]), so we show some examples below ... 

We can also permit a = b, i.e. we can distinguish two classes of a-gons in the main problem. This is similar to (R, <7)-polycycles considered before, where faces were partitioned into holes and proper ones. 

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