Chemistry of Period 2 Oscillators (Part 2 of a series of articles under the general name "Oscichemistry"). The previous article, "Barber-chemistry", considered oscillators compatible with a barberpole. Here we shall consider similar structures based on other period-2 wicks, and we shall also consider linkages that are not implied directly by any simple wicks. 1. Starting all over again, we shall now consider oscillators compatible with the following wick due to Robert Wainwright. Just as the basic element of a barberpole is the obo-spark, this wick is composed of domino sparks. Therefore we shall refer to it here as "Wainwright's domino wick", and the corresponding valency will be a "domino-valency". #C Robert Wainwright's domino wick x = 17, y = 16, rule = B3/S23 oo$o$bobo$$3oboo$$6boo$$8boo$$10boo3bo$15bo$12boobbo$$14bobo$15boo! With domino-radicals it is possible to construct a chemistry similar in many respects to barber-chemistry. For example, here are several oscillators consisting of pairs of monovalent domino-radicals: #C Oscillators formed from monovalent domino-radicals x = 161, y = 34, rule = B3/S23 oo23boo28boo23boo31boo26boo$o24bo29bo24bo32bo27bo$bobo22bobo27bobo22bo bo30bobo25bobo$$3oboo19b3oboo24b3oboo19b3oboo27b3oboo22b3oboo$$6boo23b oo28boo23boo31boo5boo19boo$96bo28bobo$8boo23boo28boo4boo17boo5bobo23b oobobbobo19boo$68bobo23bobo29bobobo25bobo$10boo3bo19boo3boo23boo23boob obbobo24bo4bobo20boobobbo$15bo25bo26bo26bobobo25bobobo25boboboo$12boo bbo20boobo22b3o4bo21bo4bobo25bobo25bobobo$63bo6bo23bobobo27boo27bobobo $14bobo22b3o24bobobo23bobo60bo$15boo51bo26boo6$62bo$60bobobo$60bo6bo 19boo$60bo4b3o19bobo28boo$5boo55bo22bobobo8bo19bobo5boo$5bobo3bo52boo 4boo12bobo4bo5bobo16bobobbo3bobo18bobo$11bo24b3o3boo16bobo6bobo12bobob o7bobo16bobobo5bobobo17bobbo5bo$5bobboobbo30bo16boo4boo17bobobboboobo bbobo14bobo4boo4bobo13boobobo4bobobo$6bo30boboobo26bo17bobo7bobobo14bo bobo5bobobo16boboboobobo$6bo3bobo23bo27b3o4bo14bobo5bo4bobo16bobo3bobb obo15bobobo4boboboo$11boo23boo3b3o20bo6bo15bo8bobobo17boo5bobo19bo5bo bbo$67bobobo24bobo27boo27bobo$69bo27boo! Among these we see radicals that include barber-valencies as well as domino-valencies. That is, at this stage we are already dealing with diradicals which include one valency that has the nature of a barberpole, and another with the nature of a domino. An examination of P2 oscillators leads us to the conclusion that the majority can be decomposed into small parts (radicals) connected with each other by valencies of various natures. Such decomposition includes an element of arbitrariness in many cases. It is generally more convenient to define as radicals those parts of an oscillator that are least connected to other parts (separated by an interval of empty space, for example, or by a narrow "neck" of live cells in a particular generation). There should also be examples of ways to connect the remainder of the subdivided oscillator to other radicals; that is, there should be multiple radicals with valencies of the given nature. Examples of divalent domino-radicals and some homogeneous polymerisations -- i.e., wicks and self-contained figures -- are shown below: #C Divalent domino-radicals, oscillators and wicks x = 145, y = 140, rule = B3/S23 42boo$42bo$43bobo$$42b3oboob3o$bboo$bbo45bobo$3bobo$50bobo$bb3oboo$49b 3oboob3o$8boo$55bobo$10boo$57bobo$12boob3o$56b3oboob3o$14bobo$62bobo$ 16bobo$64bobo$15b3oboo$63b3oboob3o$21boo$69bobo$23boo$71bobo$25boob3o$ 70b3oboob3o$27bobo$30bo45bobo$29boo48bo$78boo18$93bo$93bo3boo$91bobbo 3bo$78bo12bob4o$78bo3boo7bobbo$76bobbo3bo13boob3o$63bo12bob4o10bobo36b o$63bo3boo7bobbo7boo3bobbo3bobo29bo3boo$61bobbo3bo13boo3bobbo3boo6bo 26bobbo3bo$48bo12bob4o10bobo8bobo10boo26bob4o$48bo3boo7bobbo7boo3bobbo 3boo43bobbo$46bobbo3bo13boo3bobbo3boo7bobbo43boo$46bob4o10bobo8bobo10b 4obo38bobo7b3o$46bobbo7boo3bobbo3boo13bo3bobbo38bobbo3boo$oo18bobboo 27boo3bobbo3boo7bobbo7boo3bo34boobbo3boo7boboo$o19bobobo10boo10bobo8bo bo10b4obo12bo34bo3bo7bobob3o$bobo16bo14bo6boo3bobbo3boo13bo3bobbo49bo 9boo3bo$22bo13bobo3bobbo3boo7bobbo7boo3bo51bo3boo9bo$3oboo14bo22bobo 10b4obo12bo50b3obobo7bo3bo$20bo14b3oboo13bo3bobbo61boobo7boo3bobboo$6b oo10bo24bobbo7boo3bo69boo3bobbo$13boo3bo22b4obo12bo65b3o7bobo$8boo3bo bbo22bo3bobbo84boo$14bobo22boo3bo90bobbo$10boo32bo88b4obo$14bobbo113bo 3bobbo$12b4obo113boo3bo$10bo3bobbo66b3obboo45bo$10boo3bo72bobo$15bo66b oobo$71b3obboo6b3oboo3bo$75bobo7bo7bo$69boobo7boo3bo4boobbo$58b3obboo 6b3oboo3bobbo3bo$62bobo7bo8bobobboo4bobo$56boobo7boo3bo4boo14boo$45b3o bboo6b3oboo3bobbo3bo6bobbo$49bobo7bo8bobobboo4b4obo$43boobo7boo3bo4boo 11bo3bobbo$45b3oboo3bobbo3bo6bobbo5boo3bo$34boo10bo8bobobboo4b4obo10bo $34bo6boo3bo4boo11bo3bobbo$35bobo3bobbo3bo6bobbo5boo3bo$42bobobboo4b4o bo10bo$34b3oboo11bo3bobbo$42bobbo5boo3bo$40b4obo10bo$38bo3bobbo$38boo 3bo$43bo16$77boo$78bo$77bo4boo$6boobbo55boo9boo4bo$6bobobo56bo12boo$ 10bo55bo4boobb4o$8bo46boo9boo4bobbo6boob3o$bboo6bo45bo12boo6bo$bbo7bo 44bo4boobb4o9bo6bobo$3bobo6bo31boo9boo4bobbo6boo6bo7bo$12bo32bo12boo6b o9b4o6boo$bb3oboo6bo29bo4boobb4o9bo6boo$14bo29boo4bobbo6boo6bobbo4boo$ 8boo6bo30boo6bo9b4obboo4bo$16bo17boo6b4o9bo6boo12bo$10boo6bo15bo7bo6b oo6bobbo4boo9boo$15b4o16bobo6bo9b4obboo4bo$12boo30bo6boo12bo$10bo4boo 17b3oboo6bobbo4boo9boo$10boo4bo26b4obboo4bo$15bo24boo12bo$15boo21bo4b oo9boo$38boo4bo$43bo$43boo! The domino-wick differs from a barberpole in that it has no mirror-symmetry (a barberpole has glide-reflective symmetry). Therefore domino-radicals have many more combinations of possible valency orientations. There are seven distinct varieties of divalent domino-radical, where for barber-radicals there were only three. For higher valencies this difference is even larger. I did not attempt to deduce, for all n, the common formula for the number of varieties of n-valent domino-radicals. There are several reasons for this. First, the problem is much more difficult than its equivalent for symmetric linkages. Second, any such formula will be true only for covalent linkages -- and later we shall consider polar linkages as well. Third and most important, multivalent radicals with linkages of a single type do not play a particularly important role in the structure of oscillators. Radicals having valencies of multiple types are more important -- simply because such radicals are smaller, as a rule. And taking into account all possible varieties of multi-type linkages is impossible. Of the seven types of divalent domino-radicals, three are very similar to three types of divalent barber-radicals. In them, one linkage can be mapped onto the other by a 90- or 180-degree rotation, or a translation (for 0-degree "rotations"). These can be considered valencies of types I, L and U -- everything stated about barber-radicals of these types also holds for domino-radicals The remaining four linkage types for divalent domino-radicals involve a reflection across an orthogonal or diagonal axis, along with an associated translation. Some of these 'reflective linkage types' are reminiscent of type-L radicals, but they cannot form homogeneous self-contained figures. The only homogeneous polymerisations of reflective radicals are wicks. However, it is possible to create inhomogeneous self-contained oscillators using a combination of radicals of different reflective types (see domino.lif, item 5). Some divalent domino-radicals also include barber-valencies closed by preblocks, so they are really radicals of a higher valency, but with some linkages of a different nature. Quadrivalent variants that can yield agars are most interesting; some of these are shown in an associated file (items 2, 3, and 6 in domino.lif). Quadrivalent radicals are also possible where all four valencies are domino-type. Of these, the following radical is most interesting. In the densest polymerisation the agar similar to Don Woods' 'squaredance' is formed. In the absence of another name, we shall refer to it as his rock-and-roll agar (in my opinion rock-and-roll is more "square" than a squaredance -- a common rock-and-roll chord sequence is referred to as "square", or "quadrate", by Russian rock musicians). By the way, can anyone provide a stabilization of this agar perimeter for larger squares? #C Quadrivalent domino-type radical and agars x = 130, y = 43, rule = B3/S23 18boboo32booboo52boboo$16boo3bo26boo5bobobboo47boo3bo$oo18bo28bo3bo4bo bobbo49bo5boo$obo15bo30bob3oboobo3boo39boobbo3bo8bo$18bo31bo5bobbo43bo bobo5bo3bobo$obboo11bo35b3oboobbo46bo5bo$bo14bo34boobbobo8boo29boobbo 3bo9boob3o$bo3boo7bo30bo4bo3bo5boo3bobbo28bobobo5bo3boo$14bo29bobo3bo 5boo7boobo32bo5bo6bo7boo$7boo3bo29b3obo5boo7b3obobboo21boobbo3bo9boo3b o8bo$12bo28bo6boo7boo4bobbo3bo20bobobo5bo3boo9bo3bobo$9boo30bobobo7boo 5bo3bo3boo25bo5bo6bo7bo$13boo27b3o4boo5bo3bo32bo9boo3bo9boob3o$11bo34b o5bo3bo5boo3bo8bo18bo3boo9bo3boo$11bo3boo20boo5boobbo3bo5boo5bobo6b3o 18bo6bo7bo6bo7boo$9bo28bo5bo3bo5boo5bo3bo3boobbo15boo6boo3bo9boo3bo8bo $9bo7boo3bo13bo5bobo5boo5bo3bo9bobbo14boboboo9bo3boo9bo3bobo$7bo14bo 13boobbobo3boo5bo3bo5boo3bobboobo21bo7bo6bo7bo$7bo11boobbo15bo9bo3bo5b oo5bobo3boboo13bobbo3bo9boo3bo9boob3o$5bo29boboo6bo3bo5boo5bo3bo3bobo 4bo12bo7bo3boo9bo3boo$5bo15bobo11boo3boobobo5boo5bo3bo8bobboboo12bo7bo 6bo7bo6bo7bo$3bo18boo14b3obbo3boo5bo3bo5boo3bobb3o25boo3bo9boo3bo7bo$ bbo3boo27boobobbo8bo3bo5boo5boboboo3boo14b3oboo9bo3boo9bo3bobbo$bboobo 29bo4bobo3bo3bo5boo5bo3bo6boobo21bo7bo6bo7bo$37boobo3bobo5boo5bo3bo9bo 19bobo3bo9boo3bo9boobobo$38boboobbo3boo5bo3bo5boo3bobobboo15bo8bo3boo 9bo3boo6boo$38bobbo9bo3bo5boo5bobo5bo15boo7bo6bo7bo6bo$39bobboo3bo3bo 5boo5bo3bo5bo28boo3bo9boo3bo$36b3o6bobo5boo5bo3bobboo5boo19b3oboo9bo3b oo9bo$36bo8bo3boo5bo3bo5bo35bo7bo6bo5bo$52bo3bo5boo4b3o25bobo3bo9boo3b o5bobobo$43boo3bo3bo5boo7bobobo23bo8bo3boo9bo3bobboo$42bo3bobbo4boo7b oo6bo23boo7bo6bo5bo$43boobbob3o7boo5bob3o35boo3bo5bobobo$44boboo7boo5b o3bobo29b3oboo9bo3bobboo$44bobbo3boo5bo3bo4bo37bo5bo$45boo8bobobboo37b obo3bo5bobobo$52bobboob3o37bo8bo3bobboo$53bobbo5bo35boo5bo$49boo3boboo b3obo40bo3boo$49bobbobo4bo3bo40boobo$51boobbobo5boo$54booboo! 2. Another wick of Robert Wainwright's also consists of lines of domino sparks in one phase. But the other phase of the wick consists of obo sparks, similar to a barberpole. I shall name his 'obo-do wick'. Here "do" is simultaneously the initial syllable of "domino" and an abbreviation of the words "double o", corresponding to the RLE encoding of a domino spark. #C Robert Wainwright's obo-do wick x = 17, y = 16, rule = B3/S23 bbobo$obo$o4boo$o3bo$bbo$4bo$4bobo$6bobo$8bobo$10bobo$12bo$14bo$12bo3b o$10boo4bo$14bobo$12bobo! Obo-do linkage is similar in many respects to domino linkage. It also is nonsymmetric, and thus yields 7 types of divalent obo-do radicals. Among the quadrivalent obo-do radicals the following radical is most interesting. Its polymerisation also yields a rock-and-roll agar. Thus, this agar has a dual nature: it can either be decomposed into cruciform radicals connected by domino-linkages, or alternatively into square radicals connected by obo-do linkages. However, sparser agars formed from obo-do radicals differ from their domino analogs. #C Quadrivalent obo-do radical, and second and third agars in series #C (rock-and-roll agar is not shown) x = 143, y = 51, rule = B3/S23 48bo$46bobo65b3o$18b3o29bo53bobo$45b5o9bo42bobo8bobboo$17bobboo29bo5bo bo42bo4boo13b3o$bbobo41b5o4bobobo4boo36bo3bo4boobobbo$obo12boobobbo27b obobo11bo38bo10bo5bobboo$o4boo12bo27bo6bo3bo3bobo5boo3bo5bo24bo3boo3bo 14b3o$o3bo9boo3bo29boboo5bobo10bo3bo3bobobo22bobo10boobobbo$bbo42boo8b oo3bo3bo3bobo3bobbobobo26bo3bo10bo5bobboo$4bo8boo32bobo3bo10bobo12bobo boo16bobo8bobo3boo3bo14b3o$4bobo46bo3bo3boo3bo3bo3boboboobo17bobo7boo 3bobo10boobobbo$6bobo3boo36boo5bobo10bobo6bobbo16bo4boo10bo3bo10bo5bo bboo$8bobo34boo7boo3bo3bo3boo3bo3bobobo18bo3bo4boo10bobo3boo3bo$10bo3b o30boboboo12bobo10bo24bo15boo3bobo10boobobbo$14bobo36boo5boo3bo3bo3boo 3bo24bo3boo15bo3bo10bo$11boo3bobo28bo3bo17bobo5bo25bobo10boo10bobo3boo 3bo$18bobo30bo3bo3boo5boo3bo3bo3bo25bo3bo15boo3bobo$10boo8bo27boo5bobo 17bo22bobo8bobo3boo15bo3bo$22bo20boo7boo3bo3bo3boo5boo5boo15bobo7boo3b obo10boo10bobo$5bo3boo9bo3bo18boboboo12bobo12boo18bo4boo10bo3bo15boo3b o$5bo12boo4bo26boo5boo3bo3bo3boo6b3o14bo3bo4boo10bobo3boo15bo$3bobbob oo12bobo20bo3bo17bobo5boo21bo15boo3bobo10boo4bo3bo$20bobo26bo3bo3boo5b oo3bo3bo26bo3boo15bo3bo10boo4bo$3boobbo38boo5bobo17bo3bo22bobo10boo10b obo3boo7bobo$41b3o6boo3bo3bo3boo5boo30bo3bo15boo3bobo8bobo$4b3o38boo 12bobo12boobobo15bobo8bobo3boo15bo3bo$42boo5boo5boo3bo3bo3boo7boo13bob o7boo3bobo10boo10bobo$47bo17bobo5boo18bo4boo10bo3bo15boo3bo$43bo3bo3bo 3boo5boo3bo3bo21bo3bo4boo10bobo3boo15bo$45bo5bobo17bo3bo19bo15boo3bobo 10boo4bo3bo$44bo3boo3bo3bo3boo5boo27bo3boo15bo3bo10boo4bo$46bo10bobo 12boobobo19bobo10boo10bobo3boo7bobo$42bobobo3bo3boo3bo3bo3boo7boo21bo 3bo15boo3bobo8bobo$40bobbo6bobo10bobo5boo30bobo3boo15bo3bo$41boboobobo 3bo3bo3boo3bo3bo26bo3boo3bobo10boo10bobo$38boobobo12bobo10bo3bobo20bo 10bo3bo15boo3bo$41bobobobbo3bobo3bo3bo3boo8boo16bobboboo10bobo3boo15bo $39bobobo3bo3bo10bobo5boobo30bo3boo3bobo10boo4bo3bo$41bo5bo3boo5bobo3b o3bo6bo18boobbo5bo10bo3bo10boo4bo$57bo11bobobo28bobboboo10bobo3boo7bob o$57boo4bobobo4b5o18b3o14bo3boo3bobo8bobo$63bobo5bo30boobbo5bo10bo3bo$ 63bo9b5o32bobboboo10bobo$72bo30b3o14bo3boo3bo$74bobo33boobbo5bo10bo$ 74bo43bobboboo4bo3bo$111b3o13boo4bo$118boobbo8bobo$129bobo$119b3o! 3. The following wick, found by Dean Hickerson, consists of oblique triplets. Triplets are connected with each other by a single live cell; therefore we shall use the term "dot valency" to refer to the valency connecting triplets with each other. #C Dean Hickerson's oblique-triplet wick x = 18, y = 25, rule = B3/S23 3bo$bobo$5bo$6o$$b3o$3bo$bbo$$3boo$5bo$$6boo$8bo$$9boo5boo$11bo3bobo$ 13bo$13bobbo$9b3o$15boo$9bo$8bo6bobo$8boobobobboo$13bo! The domino part of one triplet can be linked with the single dot of the next triplet in two mirror-image orientations, so these radicals can be connected without worrying about chirality (left- or right-handedness). So for dot-valency, as for barber-valency, there are only three types of diradicals instead of seven: U, L and I. However, in contrast to barber-valency, the direction of linkage is important: dot linkage has polarity. We shall say that the dot extremity of an oblique triplet has positive polarity, and the domino extremity is negative. The polarity of a dot linkage reverses every generation. We shall name this type of linkage a 'semipolar' linkage to distinguish it from a 'full polar' linkage in which polarity does not vary with time. With a semipolar symmetric linkage of this type, there are six types of diradicals instead of three: for each divalent type, variants may have valencies with matching polarity or opposite polarity. Since the polarities change in synchrony with each other, the type of the diradical does not vary with time -- so the type can safely be used to predict the behavior of the radical in homogeneous polymerisations. For example, type-I diradicals with valencies of opposite polarity can be connected immediately to copies of themselves, forming wicks. But type-I diradicals with valencies of matching polarity cannot be connected in this way; successive radicals must be in opposite phases for the connections to work. By contrast, alternation of phases is not generally necessary for barber- and domino-radicals (I did it sometimes to derive a tighter joint). There are some small divalent dot radicals that can form wicks that are just as simple as a barberpole, domino or obo-do-wick. These wicks are connected by the same kind of dot-valency as Dean Hickerson's triplet wick. Thus their basic components can be classified as I-type dot-valency diradicals with valencies of opposite polarities, just like the oblique triplet. These wicks also share other characteristics of oblique-triplet wicks. #C Wicks formed from type-I dot-valency diradicals x = 79, y = 87, rule = B3/S23 43bo$41bobo$45bo$40b6o$$41b3o$43bo$42bo$$3bo39boo$bobo$5bo38bo$6o$45b oo$b3o$3bo42bo$bbo$47boo$3boo$5bo42bo$$6boo41boo$8bo$50bo$9boo$11bo39b oo$$12boo38bo$$13bo39boo$$14boo38bo$16bo$55boo$17boo$19bo36bo$27boo$ 20boo5bobo27boo$22boboobo$25boboboo27bo$24bobbo$26bobo29bobo$58boo$61b o$56b5o$56bo4bo$57bobo$59bo4$47bo$45bobo$49bo$3bo40b6o$bobo$5bo39b3o$ 6o41bo$46bo$b3o$3bo43boo$bbo46bo$48bobo$3boo$5bo45boo$53bo$6boo44bobo$ 8bo$55boo$9boo46bo$11bo44bobo$$12boo45boo$14bo46bo$13bobo44bobo$$16boo 45boo$18bo46bo$64bobo$19boo$21bo45boo6boo$29boo38bo5bobo$22boo5bobo36b oboboobo$24boboobo43boboboo$27boboboo39bobbo$26bobbo44bobo$28bobo! We shall name one of these wicks (due to Robert Wainwright) the 'dash-and-dot wick' because of its appearance. The valency connecting this wick is the same familiar dot-valency discussed above, and almost all dot radicals found for a triplet wick will work perfectly with the dash-and-dot wick. In the associated files of search-program results (triplet.lif and dashdot.lif) many radicals will appear to consist of smaller elements. I did not discard such composite radicals, as they show possible methods of constructing oscillators from smaller elements. Because of the semipolarity of dot linkage there are, for example, 4 subtypes of type-X quadrivalent radicals. The smallest such radical looks like a diagonal two-bit spark: two opposing valencies have positive polarity, and the other two are negative. The densest polymerisation of this radical is the known agar consisting of diagonal two-bit sparks. A sparser variant is also shown below. #C Agars based on diagonal two-bit sparks x = 161, y = 56, rule = B3/S23 110boobo6boobo6boobo6boobo6boobo$110bo3boo4bo3boo4bo3boo4bo3boo4bo3boo $54bo7bo7bo7bo32bo9bo9bo9bo9bo$48boo4bobo5bobo5bobo5bobo32bo9bo9bo9bo 9bo$45bobbo3bo7bo7bo7bo7boo25bo3bo5bo3bo5bo3bo5bo3bo5bo3bo$43bobbobo4b 5o3b5o3b5o3b5obo22bobboo4bo3boo4bo3boo4bo3boo4bo3boo4bo3boo$43bobboboo bbo7bo7bo7bo9bo19bo9bo9bo9bo9bo9bobobo$44bo3bo5bobo5bobo5bobo5bobo3boo 19bobbo9bo9bo9bo9bo$48bo3bo3bo3bo3bo3bo3bo3bo3bo36bo9bo9bo9bo9bobbo$bb o19bo21bo5bo3bo3bo3bo3bo3bo3bo3bo3b6o17bobobo9bo9bo9bo9bo9bo$bbobo17bo bo17boboboo3bo3bo3bo3bo3bo3bo3bo3bo4bobbo17boo3bo4boo3bo4boo3bo4boo3bo 4boo3bo4boobbo$o4bo14bo23bo4bo3bo3bo3bo3bo3bo3bo3bo3bo28bo3bo5bo3bo5bo 3bo5bo3bo5bo3bo$b5o14b6o15boobobobo3bo3bo3bo3bo3bo3bo3bo3bo31bo9bo9bo 9bo9bo$o43bo5bo3bo3bo3bo3bo3bo3bo3bo3bobobo26bo9bo9bo9bo9bo$bboo18b3o 18bobobo3bo3bo3bo3bo3bo3bo3bo3bo5bo25bo3bo5bo3bo5bo3bo5bo3bo5bo3bo$bob o18bo26bo3bo3bo3bo3bo3bo3bo3bo3boboboboo17bobboo4bo3boo4bo3boo4bo3boo 4bo3boo4bo3boo$23bo24bo3bo3bo3bo3bo3bo3bo3bo3bo4bo20bo9bo9bo9bo9bo9bob obo$3bo40bo5bo3bo3bo3bo3bo3bo3bo3bo3boobobo17bobbo9bo9bo9bo9bo$4bo16b oo19boboboo3bo3bo3bo3bo3bo3bo3bo3bo5bo31bo9bo9bo9bo9bobbo$4bo15bo23bo 4bo3bo3bo3bo3bo3bo3bo3bo3bo23bobobo9bo9bo9bo9bo9bo$6bo34boobobobo3bo3b o3bo3bo3bo3bo3bo3bo24boo3bo4boo3bo4boo3bo4boo3bo4boo3bo4boobbo$7bo10b oo24bo5bo3bo3bo3bo3bo3bo3bo3bo3bobobo23bo3bo5bo3bo5bo3bo5bo3bo5bo3bo$ 7bo9bo25bobobo3bo3bo3bo3bo3bo3bo3bo3bo5bo26bo9bo9bo9bo9bo$9bo39bo3bo3b o3bo3bo3bo3bo3bo3boboboboo24bo9bo9bo9bo9bo$10bo4boo31bo3bo3bo3bo3bo3bo 3bo3bo3bo4bo25bo3bo5bo3bo5bo3bo5bo3bo5bo3bo$10bo3bo29bo5bo3bo3bo3bo3bo 3bo3bo3bo3boobobo18bobboo4bo3boo4bo3boo4bo3boo4bo3boo4bo3boo$12bo29bob oboo3bo3bo3bo3bo3bo3bo3bo3bo5bo20bo9bo9bo9bo9bo9bobobo$13bo30bo4bo3bo 3bo3bo3bo3bo3bo3bo3bo23bobbo9bo9bo9bo9bo$11bo3bo25boobobobo3bo3bo3bo3b o3bo3bo3bo3bo36bo9bo9bo9bo9bobbo$9boo4bo28bo5bo3bo3bo3bo3bo3bo3bo3bo3b obobo18bobobo9bo9bo9bo9bo9bo$16bo26bobobo3bo3bo3bo3bo3bo3bo3bo3bo5bo 19boo3bo4boo3bo4boo3bo4boo3bo4boo3bo4boobbo$8bo9bo30bo3bo3bo3bo3bo3bo 3bo3bo3boboboboo21bo3bo5bo3bo5bo3bo5bo3bo5bo3bo$6boo10bo29bo3bo3bo3bo 3bo3bo3bo3bo3bo4bo26bo9bo9bo9bo9bo$19bo24bo5bo3bo3bo3bo3bo3bo3bo3bo3b oobobo25bo9bo9bo9bo9bo$5bo15bo20boboboo3bo3bo3bo3bo3bo3bo3bo3bo5bo25bo 3bo5bo3bo5bo3bo5bo3bo5bo3bo$3boo16bo22bo4bo3bo3bo3bo3bo3bo3bo3bo3bo24b obboo4bo3boo4bo3boo4bo3boo4bo3boo4bo3boo$22bo18boobobobo3bo3bo3bo3bo3b o3bo3bo3bo25bo9bo9bo9bo9bo9bobobo$bbo41bo5bo3bo3bo3bo3bo3bo3bo3bo3bobo bo18bobbo9bo9bo9bo9bo$3bo18bobo18bobobo3bo3bo3bo3bo3bo3bo3bo3bo5bo31bo 9bo9bo9bo9bobbo$b3o18boo25bo3bo3bo3bo3bo3bo3bo3bo3boboboboo16bobobo9bo 9bo9bo9bo9bo$25bo22bo3bo3bo3bo3bo3bo3bo3bo3bo4bo19boo3bo4boo3bo4boo3bo 4boo3bo4boo3bo4boobbo$6o14b5o17bobbo4bo3bo3bo3bo3bo3bo3bo3bo3boobobo 22bo3bo5bo3bo5bo3bo5bo3bo5bo3bo$5bo14bo4bo16b6o3bo3bo3bo3bo3bo3bo3bo3b o5bo26bo9bo9bo9bo9bo$bobo17bobo25bo3bo3bo3bo3bo3bo3bo3bo3bo31bo9bo9bo 9bo9bo$3bo19bo20boo3bobo5bobo5bobo5bobo5bo3bo25bo3bo5bo3bo5bo3bo5bo3bo 5bo3bo$43bo9bo7bo7bo7bobboobobbo19bobboo4bo3boo4bo3boo4bo3boo4bo3boo4b o3boo$46bob5o3b5o3b5o3b5o4bobobbo19bo9bo9bo9bo9bo9bobobo$44boo7bo7bo7b o7bo3bobbo20bobbo9bo9bo9bo9bo$49bobo5bobo5bobo5bobo4boo35bo9bo9bo9bo9b obbo$51bo7bo7bo7bo29bobobo9bo9bo9bo9bo9bo$105boo3bo4boo3bo4boo3bo4boo 3bo4boo3bo4boobbo$110bo3bo5bo3bo5bo3bo5bo3bo5bo3bo$112bo9bo9bo9bo9bo$ 114bo9bo9bo9bo9bo$110boo3bo4boo3bo4boo3bo4boo3bo4boo3bo$112boboo6boboo 6boboo6boboo6boboo! Segments of a barberpole or obo-do wick of any length are also quadrivalent dot radicals. In particular, the domino-spark is also a quadrivalent dot radical. Therefore, for example, the rock-and-roll agar can be considered to be a polymerisation of domino-spark radicals -- a much smaller basic unit than the radicals considered earlier, which consist of four dominoes. Valencies can be identified in other period-2 wicks without any great difficulty. Next we shall consider another mode of detection of valencies, starting with any simple oscillator rather than a wick. 4. We shall consider, for example, a beacon. A beacon consist of two "poorly connected" halves (in one phase they make separate islands, and in the other they are connected at only one point). Can we find other "beacon-radicals" -- that is,. can we find oscillators consisting of a half-beacon connected via a blinking diagonal spark to some other structure (different from a half-beacon)? Here are examples of such oscillators: #C Sample beacon-radicals x = 33, y = 11, rule = B3/S23 4booboo$4bo4bo15boobbo$oo3boo4bo13bo3b3o$obboboob4o9boo3b3o3bo$6bo14bo bbobobooboo$bboobobb4o$3b3obbobbo11boob3obboo$3o21boo3bobbo$obbo17b3o bboobobo$boo18bobbobboboboo$24boobbo! And there are also examples of divalent beacon-radicals, though unfortunately the majority of them can not form homogeneous wicks: #C Oscillators formed from divalent beacon-radicals x = 49, y = 15, rule = B3/S23 47boo$5bo16booboo15boo3bo$oobobo3bo12bo4bo15bo4bo$oobobbobboo7boo3boo 4bo13bob4o$bbo6boo7bobboboob4o10boobobo$bboob5obo12bo15boobo3boo$4boo 5bo8boobobboo14booboobo$obboo3bo12b3obboo14bo3bo$oo3boobboo7b3o7boo12b oo4bo$4bo13bobbo6boo10boobo3boo$19boo19boobobo$43bob4o$43bo4bo$42boo3b o$47boo! Such a wick can be constructed from the following (known) divalent beacon-oscillator, which is simultaneously a divalent barber-oscillator -- it is based on a quadrivalent radical with two barber-valencies and two beacon-valencies. This also allows the construction of an agar: #C Quadrivalent barber- and beacon-radical, wick and agar x = 92, y = 42, rule = B3/S23 69boo$63boo5bo$63boobbobo$65boo5boo$65boobobbobo$59boo3bo4bo$23boo34bo bbo3bo4bo3boo$7boo8boo3bobo39bobboboo5bo$boo3bobo8bobbo39boboo5boobbob o$bobbo17bo3boo27boo5boobbobobboo5boo$6bo11boboo5bo27boobbobobboo5boob obbobo$bboboo14boobbobo30boo5boobobbo4bo$4boo11bobobboo5boo26boobobbo 4bo3bo4bo3boo$bobobboo8bo5boobobbobo20boo3bo4bo3bo4bobboboo5bo$o5boo8b oo3bo4bo24bobbo3bo4bobboboo5boobbobo$oo21bo4bo3boo22bobboboo5boobbobo bboo5boo$19bobobboboo5bo18boboo5boobbobobboo5boobobbobo$19boo5boobbobo 21boobbobobboo5boobobbo4bo$23bobobboo5boo14bobobboo5boobobbo4bo3bo4bo 3boo$22bo5boobobbobo13bo5boobobbo4bo3bo4bobboboo5bo$22boo3bo4bo17boo3b o4bo3bo4bobboboo5boobbobo$29bo4bo3boo17bo4bobboboo5boobbobobboo5boo$ 25bobobboboo5bo13bobobboboo5boobbobobboo5boobobbobo$25boo5boobbobo14b oo5boobbobobboo5boobobbo4bo$29bobobboo5boo14bobobboo5boobobbo4bo3bo4bo $28bo5boobobbobo13bo5boobobbo4bo3bo4bobboboo$28boo3bo4bo17boo3bo4bo3bo 4bobboboo5boo$35bo4bo3boo17bo4bobboboo5boobbobobboo$31bobobboboo5bo13b obobboboo5boobbobobboo5boo$31boo5boobbobo14boo5boobbobobboo5boobo$35bo bobboo21bobobboo5boobobbo$34bo5boobo18bo5boobobbo4bo3bobbo$34boo3bo22b oo3bo4bo3bo4bo3boo$41bobbo24bo4bobboboo$37bobo3boo20bobobboboo5boo$37b oo26boo5boobbobobboo$69bobobboo5boo$68bo5boobo$68boo3bo$75bobbo$71bobo 3boo$71boo! 5. A toad is another simple oscillator that breaks up into two islands in one of its phases. Here are some examples of divalent oscillators having toad-valency, and also wicks based on this type of linkage: #C Divalent toad-radicals and wicks x = 88, y = 52, rule = B3/S23 4booboo24booboo$bo3boboo21bo3boboo$obbobo23bobbobo$obbobo23bobbobo$bbo boboo23boboboo$4bobo26bobo4booboo$3boobobo23boobobo3boboo$5bobobbo23bo bobbobo$5bobobbo23bobobbobo$bboobo3bo21boobo3boboboo$bbooboo24booboo4b obo4booboo$39boobobo3boboo$41bobobbobo$41bobobbobo$38boobo3boboboo$38b ooboo4bobo4booboo$46boobobo3boboo$48bobobbobo$48bobobbobo$45boobo3bobo boo$45booboo4bobo$53boobobo$55bobobbo$55bobobbo$52boobo3bo$52booboo11$ 78bo7bo$74boobo3bobbobbo$8bo7bo57boo5bobbobbo$4boobo3bobbobbo46bo7bob oo5bo3bo$4boo5bobbobbo42boobo3bobbobbobb7o$bboboo5bo3bo44boo5bobbobbo bbo5bo$obbobb7o37bo7boboo5bo3bo10bo$obbobbo5bo33boobo3bobbobbobb7o$bo 10bo33boo5bobbobbobbo5bo$36bo7boboo5bo3bo10bo$32boobo3bobbobbobb7o$32b oo5bobbobbobbo5bo$30boboo5bo3bo10bo$28bobbobb7o$28bobbobbo5bo$29bo10bo ! 6. The following pattern, sent to me by Heinrich Koenig, is a perfect example of a series of complex constructions using the linkage types mentioned above, along with other period-2 valencies. #C Sample constructions using a variety of period-2 valencies x = 235, y = 188, rule = B3/S23 39bo$37bobo$41bo$36b6o$34bo$32bobo3boo$30bo9bo15bo$28bobo6bo16bobo$38b oo13bo4bo58bo$26boo25b5o16boo39bobo$25bo32bo14bobo43bo$55boo18bob3o7bo 26b6o$23boo30bobo14boobobb3o4bobo24bo$22bo52bo13bo22bo3boo$20bo34bo18b o9b6o21bo6bo$20bo33bo21bo5bo26bo5bo$19bo34bo22bo4bo3boo21bo6boo$17bo 34bo24bo3bo6bo19bo55bo$17bo33bo20boo5bo5bo78bobo$5bo10bo34bo22bo5bo5b oo11bo6boo54bo4bo$4boo8bo34bo21bo6bo3bo15boo5bo41boo14b5o$4boo6bobo32b oo23boo3bo4bo15boo47bobo9bobo$4bo5bo66bo5bo14bo4boo54bo5bo$8bobo35bo 23b6o9bo16bo46bobo3bobo6boo$bb5o37boo24bo13bo11b5o54bo8boo$bbo4bo64bob o4b3obboboo8bo4bo49bobo10bo20bo$3bobo37bo28bo7b3obo12bobo83bobo$5bo38b o39bobo12bo51bo35bo$42b3o39boo96b6o$150boo28bo$41b6o131bobo3boo$46bo 102bo26bo9bo$42bobo129bobo6bo$44bo103boo22bo11boo$171bobo$147bo23bo$ 172boo$117bo28boo27bo$115bobo57bobo$119bo28bo30bo$68boo28boo14b6o59bob o$68boo28boo12bo35boo33bo$79bo9bo20bobo3boo65bobo$66b6o5boo8bobo6b6o6b o9bo31bo36bo$66bo4bo7boo7bobo5bo4bo4bobo6bo71boboboo$64bobobo5bo3bo5bo 3bo5bobobo5bo11boo32boo40bo$62bobo3bobobobo7bobo7bobo3bobobobo86bo$50b o3bobobobo15b5o5b5o61bo$48bobobobo5bo20bo9bo59bo38bobo$16bo3bobobobo3b obobobo3bobobobo5bo4bo19bobo7bobo61boobobo31bobboo$3bo10bobobobo5bobob o5bobobo5bo5b6o21bo7bobbo65bo29bobo$bboo8bo5bo4bo4bo4bo4bo4bo44boo68bo bo7bo7bo7bo$bboo6bobo5b6o4b6o4b6o10boo104bo5bobo5bobo5bobo$bbo5bo45boo 106bobo5bobo5bobo$6bobo11boo8boo8boo122bo7bo7bo$5o15boo8boo8boo$o4bo$b obo$3bo4$149bo$149bobo$140bo6bo4bo$140bobo5b5o$138bo5bobo74bobo$144bo 5bo68bobbobbo$137boo10boo66bobbobobobbo$149boo67bobobobobobbo$71boo15b o50bo9bo66b3obobobobobobbo$6boo62bobo13bobobo128bobbobobobobo$5bobo64b ob3o7bobobo43bo6boo41bo32b4o3bobobobob3o$7bob3o57boobobb3o4bobbooboboo 40bobo45bobo16bobo21bo3boo$4boobobb3o59bo9bo5bo41bo5bobo45bo15bobbo10b 5o10b5o$7bo14boo47bo9bo6boo46bo42b6o12boobobo16bo9bo$6bo12bobbo50bo5bo 39bo9boo46bo22bob3o10b5o10b5o$8bo8bobboboboo48bo4bo39bobo53bobo3boo15b obo19boo3bo$17bobbobo51bo3bo38bo13bo41bo9bo15bob5o10b3obobobobo3b4o$8b oo8bo3bobo44boo5bo40b6o48bobo6bo19bo17bobobobobobbo$22bo48bo5bo46bo6b oo36bo11boo18b6o10bobbobobobobob3o$10bo57bo6bo3bo39boo3bobo37boobobo 49bobbobobobobo$20boo47boo3bo4bo38bo9bobo33bo36b5o15bobbobobobbo$10boo 62bo5bo40bo6bo36bo34bo22bobbobbo$20bo46b6o46boo79b5o20bobo$12bo29boo 10boo11bo13boo65bo15bobo10bo$18boo22bo10bobo13bobo11bo62bobo15boobbo8b oo19b6o$12boo29bobo23bo75bo4bo17bobo6boo18bo$18bo32bobo30boo58bob4o22b o5bo18b6o$14bo30bobo95bobo4bo21bobo$16bobo30bobo33bo57boboboo27b5o18b 3o$12bobo32bo92boobobobobo25bo4bo18bo$18bobo30bo34boo51bobobobobo29bob o22bo$10bobo32boo89bobbobobobo31bo20bo3bobo$9bo10bobo28boo34bo47bobobo bobo53bobbobobobo$9boo10boo22bo43bo42bobbobobobo55bobbobobobobo$50bo 37bo39boobobobobobo59bobbobobobobo$43boo38b3obboboo35bobobobobobo64boo bbobobobo$49boo33b3obo35bobbobobobobo70bobobobobo$44bo43bobo31bobbobob obobo73bobbobobobo$48bo39boo32bobbobobobo78bobobobobo$44bobobbo73bo3bo bo81boobobobobo$48boo15bo61bo86boboboo$42bobo20bobo146bobo4bo$63bo61b oo88bob4o$40bobo20b6o147bo4bo$40boo83bo91bobo$65b3o151bo$65bo57boo$66b o$123bo$64boo$63bo57boo42bo$163bobobo$61boo58bo39bobobobobo$57boobo98b obobobobobobo$57bo61boo36bobobobbobbobobobo$58bo99boobo5boobbobobo$ 119bo36bo3bo10boboo$53boobobo58bo39b3o12bo3bo25boo5boo$53bo3boo56bobbo 36bo17b3o28bo4bobo$54bo58bobbobobb3o32boo19bo23bo4boobobo$111bobbobobo b3o31bo3bo16boo25boo3boboboboo$49boobobo54bobbobobobo36b3o16bo3bo28bob obobobo$49bo3boo52bobbobobobobo34bo21b3o22b6o3bobobobobboo$50bo54bobbo boboboboo36boo23bo20bo10bobobobobobbo$103bobbobobobbo38bo3bo20boo22b4o 8bobobobobobbo$45boobobo50bobbobobobo42b3o20bo3bo34bobobobobbo$45bo3b oo48bobbobobobbo41bo24b4o19b5o13bobo3bo$46bo50bobbobobobo45b4o24bo17bo 20bo$98boboboboo45bo3bo20b3o19b5o19bo$45boboboo13bo19bo11b3obobo50boo 20bo3bo40b3o$45boo3bo11bobo19bobo13bobo49bo23boo18b6o$49bo16bo15bo12b 5obo52b3o21bo16bo23b6o$61b6o15b6o12bo52bo3bo16b3o18b6o23bo$45boobobo 45bobo56boo16bo3bo40b6o$45bo3boo11b3o19b3o11bo55bo19boo21b3o$46bo17bo 19bo71b3o17bo20bo19b5o$63bo21bo69bo3bo12b3o25bo20bo$45boboboo60bo45boo bo10bo3bo20bo3bobo13b5o$45boo3bo12boboboo11boobobo25bobo42bobobobboo5b oboo21bobbobobobo$49bo13boo3bo5bo5bo3boo23bo4bo43bobobobobbobbobobo20b obbobobobobo8b4o$67bo5bobo5bo28b5o45bobobobobobobo24bobbobobobobo10bo$ 45boobobo22bobo33bo52bobobobobo29boobbobobobo3b6o$45bo3boo16boboboobob oobobo29boo34bo16bobobo35bobobobobo$46bo20boo3bobobo3boo28bobo32bobo 18bo38boobobobo3boo$71bobbobbo66bo4bo58boboboo4bo$45boboboo22bobo36bo 31b5o59bobo4bo$45boo3bo60bo3boo32bo59boo5boo$49bo61boobobo29boo$146bob o$45boobobo65bo6bo$45bo3boo64bo3boobobo21bo$46bo68boobobobobobbo14boo 3bo$120bobobobobo13boboboo$44boo76bobobobobbo39bobo$43bo80bobobobobo9b o29bobbo$34boo90bobobobobbooboo3bo25boobobo3boo$33bobo5boo85bobobobobo boboboo28boboobobo$35boboobo45bo43bobobobobo31bobo$32boobobo48bobo43bo bobo34boo6bo$35bobbo32boo13bobobo43bo43bo3boo$34bobo33bobo105boobobo$ 88bobobo$68bobo112bo$87bobobbobo55bo31bo$66bobo79bobo31b3o$86bobo5bobo 55bo$64bobo80b6o29b4o$60bo24bobo8bobo82bo$60bobobo32boo49b3o31b5o$60bo 19boobbobo63bo$62bo17bo68bo32b4o$60bobbo17bobobo95bo$61boobo84boboboo 27b3o$60bobbo19bobobo61boo3bo$62bo22bobo65bo26b4o$61bo25bo73bo17bo$ 153boboboobobo17b3o$56boobobo91boo3bobobobbo$56bo3boo95bobbobobobo11b 4o$57bo101bobbobobobbo7bo$46bo114bobbobobobo5bob3o$46bobbooboobobo105b obbobobobbobbobo$45bobbobobo3boo107bobbobobobobobbo$50bobbo113bobbobob obbo$45bobobobo117bobbobbo$45boo124bobo! Nicolay Beluchenko, April, 2004.