Barber-Chemistry (Part 1 of a series of articles under the general name "Oscichemistry"). I have carried out a search for p2 oscillators that can be joined to a barberpole. Many of the resulting oscillators are widely known. However, the search unexpectedly yielded so many results that it was necessary to divide them into groups to classify them. A top-level classification is the number of barberpoles that can be attached to an oscillator. By analogy to chemistry, it is convenient to name this the "valency" of the oscillator. Similar results can be obtained by substituting any other wick for a barberpole, so we shall speak here about barber-valency, i.e. the valency with respect to a barberpole. Thus we shall name all oscillators considered in the given work barber-active, as opposed to other oscillators to which no barberpoles can be attached. Inside each group of barber-active oscillators of a given valency, we shall define subgroups according to geometrical details analogous to those found in stereochemistry. The classification of barber-active oscillators will be based on structures which we shall name barber-radicals (another term from chemistry). Radicals in various combinations yield a huge number of oscillators, wicks and agars. Here we shall give only the minimum form of each radical, in which all valencies are closed by barberpole terminators (i.e., preblocks) -- along with some the most typical homogeneous forms, consisting of further copies of the same radical connected with each other, and (in most cases) preblocks to close any remaining valencies. The oscillators shown in the examples are not new discoveries, as a rule -- they are necessary only for understanding the text. However, since they were all generated by the WinLifeSearch (WLS) program, they do include occasional new patterns. To produce a minimal stamp collection, the majority of the examples should be removed, leaving only their respective minimal cases. 1. Monovalent barber-active oscillators: These are oscillators to which a single barber-radical can be joined. Such radicals can serve as the terminations of a barberpole and other wicks and the agars composed from barber-radicals of higher valencies. Many such wicks and agars will be shown later. Some oscillators with a barber-valency of 1 are shown in the following diagram; the minimum form and one (for glide-symmetric solutions) or two homogeneous forms are given. #C Monovalent P2 radicals and closed oscillators x = 103, y = 112, rule = B3/S23 28boobo13boo$7boo19bo3boo11bobo$7bo3booboo13bo19bo17boobb3o14boo$8bobo boboo15bo14bobbobo15bo20bobo$12bo16bobbo14bobobobo14bobobboo18bobbo$7b 3oboo15boboobo12bobobobo35bobb6o$29bo3bo14bobobo14b3obobbo14bo$30bobo 17bo21bo16bo3boo$29booboo38bo22bo$92bo$93boo5$24booboo13bo$7bo17bobo 12bobobo$3boobbo16bo3bo11bobobobo15bo21boo$3bobobbo15boboobo9bobobobo 16bo23bo$oobo21bobbo12bobobbo13bobbob3o15bo$ooboobobo18bo15bo40boo3bo$ 11booboo9bo19bobo12boobbobo22bo$8boboboboo9bobobobo17bo32b6obbo$12bo 12bo6boo12bobbobo9b3obbobobb3o8bobbo$7b3oboo16bo17bobobobo34bobo$31bo 14bobobobo15bobobboo18bobbo$29bobbo15bobobo35bobb6o$28boboobo16bo16b3o bobbo14bo$29bo3bo38bo16bo3boo$30bobo39bo22bo$29booboo58bo$93boo4$20bob oo18bo$5boo13boobbob3o13bobo$5boo16bo16bobbo18b3obbo17boo$20boobobobo 13boobbo21bo17bo$3b4o13bobobo15bobboo16boobbobbo12b3obo$3bobbo16bo3bob obo10bobbo34b3obboo$4bo27boo7bobo17bobbobobobb3o11bo$6bo22bo13bobobo 15bo22bo$bboo27bo17bo13bo4bobobboo9bo$4bobobo20bobbo13bobbobo32bobobo$ 11booboo12boboobo13bobobobo13b3obobbo9bo$8boboboboo13bo3bo12bobobobo 19bo15bobo$12bo17bobo15bobobo19bo20bobbo$7b3oboo16booboo16bo37bobb6o$ 89bo$89bo3boo$95bo$92bo$93boo15$12bo23boob3o31bo$7boobobobo21bo31boobo bo19boobbo$7bo6bo22bo3bo26bo6boo16bobobo$8bo5bo54bo7bo19bo$12bo22boobb oboboo31bobo16bo$6boo3bo25bo7bo21boo20boo5bobb3o$11bo27bo4bobo32boo8bo bobo3bo$6bobbo31bo26bo24bobo3boo$8bo31bo6boo21bo8bo10boo$8bo32boo26bo 7bo19bobbo$47bo22boo5bo12bobboobobo$43bobo30bo15bobobobo$45bo26bobo17b obobo$74bo19bo4$6bo20bo34bo$6bo20bobo32bobo$5bobbo16bo34bo$3bo26boo27b o5boo26bobo$3bo3boo15boo6bo26bo7bo23bobbo$bbo28bo25bo8bo25boboboo$o5bo 19bobo4bo34bo20booboboo$o11bo14bo7bo20boo33bo5boo$obobobobobobobo13boo bobobboo30boo18boo$bbo11bo43bobo33bobbo$8bo5bo16bo3bo23bo7bo19bobbobob obobbo$12bo47boo11bo14bo3bo5boo$6boo3bo19b3obobob3o21bobobobobobo14bo 3bo$11bo51bo11boo17bo$6bobbo27bo3bo27bo7bo12boo6bo$8bo67bobo13bobobobo $8bo26boobboboboo22boo29bo$37bo7bo33boo15bo$39bo4bobo21bo21boo5bobb3o$ 41bo28bo8bo10bobobo3bo$40bo6boo20bo7bo16bobo3boo$41boo27boo5bo13boo$ 47bo28bo21bobbo$43bobo26bobo16bobboobobo$45bo28bo18bobobobo$93bobobo$ 95bo! 2. Divalent oscillators: There are several varieties of divalent barber-active oscillators. First of all there are oscillators whose valencies are opposite each other. We shall name these "type-I oscillators" (since the extremities of the letter 'I' are opposite each other). These oscillators can sometimes be joined to each other to form wicks. #C Divalent type-I P2 radicals and wicks x = 152, y = 99, rule = B3/S23 79boo13boo25boo$79bobo12bobo24bobo$boo10boo16boo$bobbo8bobbo14bobbo44b obboob3o6bobboob3o18bobboob3o$bbobboo7bobboo13bobboo43bo14bo26bo$80bo 3bobo8bo3bobo20bo3bobo$oobbo7boobbo13boobbo52bo$bbobbo8bo17bo53boo11bo bboob3o20bobobo$4boo10bobobo13bobob3o59bo32boo$20bo79bo3bobo23bo$16bo 3bo15bo3bo91bo$16bo87bobboob3o19bo$16bobobo15b3obobo62bo28bo$22bo21bo 60bo3bobo20bo$20bobboo17bobboo85bobobo$109bobboob3o14bo$18boobbo17boo bbo65bo25bobo$20bo21bo67bo3bobo$22bobobo17bobob3o66bo18bobboob3o$26bo 89boo19bo$22bo3bo19bo3bo86bo3bobo$22bo28bo$22bobobo19b3oboo91bobobo$ 25boo121boo$145bo$147bo$147bo$149bo$147bo$147bobobo$147bobboo15$89bo 14bo$84boobobo9boobobo$84bo6bo7bo6bo$85bo14bo$boobbo15boobbo65boo13boo $bobobo15bobobo57boo13boo$5bo19bo64bo14bo$3bo19bo60bo6bo7bo11bo$3bo19b o62boboboo9bobobobobobo$bo19bo64bo14bo11bo$bobobo15bobobobo79bo$bobboo 15bo6boo83boo$25bo79boo$27bo84bo$25bo80bo11bo$27bo80bobobobobobo$23boo 6bo76bo11bo$25bobobobo82bo$31bo88boo$29bo82boo$29bo89bo$27bo85bo11bo$ 27bobobobo81bobobobobobo$27bo6boo79bo11bo$31bo89bo$33bo93boo$31bo87boo $33bo92bo$29boo6bo82bo6bo$31bobobobo84boboboo$37bo84bo$35bo$35bo$33bo$ 33bobobobo$33bo6boo$37bo$39bo$37bo$39bo$35boo6bo$37bobobobo$43bo$41bo$ 41bo$39bo$39bobobobo$39bo6boo$43bo$45bo$43bo$45bo$41boo3bo$43boboo! In another variety of divalent oscillator, the valencies are perpendicular to each other ("type-L oscillators"). These oscillators can also form various wicks. Since these wicks change their valencies with the addition of each following radical, we shall name them "zigzag wicks", as contrasted to the "direct wicks" considered previously. It is possible to form a self-contained homogeneous pattern from four (or more) type-L oscillators; this is exactly analogous to the formation of ponds or lakes from diagonal two-bit "elements", with a pond requiring the minimum four elements. #C Divalent type-L P2 radicals, wicks, and closed oscillators x = 199, y = 60, rule = B3/S23 160bo24bo$160bo24bo$34bo11bo11bo11bo25b3o3bo56bobbob3o17bobbob3o$34bo 11bo11bo11bo31bo$33bobbob3o4bobbob3o4bobbob3o4bobbob3o13boo5boboobbo4b 3o3bo44bobobobo18bobobobo$oo88bo23bo40bo$o5boo18boo5bobobobo5bobobobo 5bobobobo5bobobobo15bobobobo3bobo5boboobbo4b3o3bo28bobobo5bobobo10bobo bo5bobo$bobobobo18bo49bo49bo28bo14boo6boo16bo$27bobobobo5bobobobo5bobo bobo5bobobobo5boo13b3obobbo5bobobobo3bobo5boboobbo4b3o3bo18bo9bo14bo9b obobo$3obobbo88bo41bo16bo13bo10bo15bo$6bo19b3obobbo4b3obobbo4b3obobbo 4b3obobbo26bo5b3obobbo5bobobobo3bobo5boboobbo17bo9bo12bo13bo$6bo25bo 11bo11bo11bo39bo44boo14bo8bo15bo$32bo11bo11bo11bo39bo5b3obobbo5bobobob o3bobo15bobobo5bobobo8bobobo9bo$120bo17boo29bo8bo16boo$120bo5b3obobbo 25bobobobo16bobo5bobobo$132bo$132bo25b3obobbo18bobobobo$164bo$164bo18b 3obobbo$189bo$189bo15$oo3bo3boo$o3bobo3bo16b3obobob3o9b3obobob3o9b3obo bob3o9b3obobob3o58b3obobob3o$bobobobobo22bo19bo19bo19bo68bo21b3obobob 3o$5bo22bobobobobo11bobobobobo11bobobobobo11bobobobobo60bobobobobo22bo $3obobob3o16bo3bobo17bobo17bobo17bobo66bobo21bobobobobo$27boo3bo3bobo 3bo3bobo3bo3bobo3bo3bobo3bo3bobo3bo3bobo3bo3bobo3bo3boo47bobo3bo3bobo 19bobo$41bobo17bobo17bobo17bobo3bo43bo19bo8bobobo3bo3bobobo$38bobobobo bo11bobobobobo11bobobobobo11bobobobobo44bobobo11bobobo6boo17boo$42bo 19bo19bo19bo48bo19bo10bo11bo$37b3obobob3o9b3obobob3o9b3obobob3o9b3obob ob3o45bo15bo10bo15bo$151bobbo13bobbo10bo11bo$152boobo11boboo8b3obo9bob 3o$151bobbo13bobbo10bo11bo$153bo15bo10bo15bo$151bo19bo10bo11bo$151bobo bo11bobobo6boo17boo$37bo7bo9bo7bo9bo7bo9bo7bo51bo19bo8bobobo3bo3bobobo $37bo3bo3bo9bo3bo3bo9bo3bo3bo9bo3bo3bo55bobo3bo3bobo19bobo$36bobbobobo bbo7bobbobobobbo7bobbobobobbo7bobbobobobbo59bobo21bobobobobo$41bo17bo 17bo17bo61bobobobobo22bo$27boo3bo3bobobobobobo3bo3bobobobobobo3bo3bobo bobobobo3bo3bobobobobobo60bo21b3obobob3o$27bo3bobo7bo7bobo7bo7bobo7bo 7bobo7bo3boo55b3obobob3o$28bobobobobo9bobobobobo9bobobobobo9bobobobobo $32bo17bo17bo17bo$27b3obobob3o7b3obobob3o7b3obobob3o7b3obobob3o! If an oscillator's two valencies are both pointed in the same direction, we shall call it a "type-U oscillator". Type-U oscillators can also form self-contained homogeneous patterns. But in this case they behave similarly to monovalent oscillators, in that two and only two identical type-U oscillators are necessary to close a contour. Wicks based on type-U oscillators are not always possible; it all depends on the relationship between the width of the oscillator in direction perpendicular to the valencies (w), and the distance between valencies (d). For a type-U oscillator to form a wick, the condition w + delta > 2d should be observed, where delta is the minimum admissible distance between parts of a wick. Routinely delta=2, but depending on the form of a radical it may be less (even becoming negative in some cases). We shall name this condition a "U-relation", since its minimal case has appeared for type U. As we shall see later, the same relation will play a significant role for radicals with higher valencies. #C Divalent type-U P2 radical, wick, and oscillators x = 30, y = 86, rule = B3/S23 8boo$8bobo$$5boo3bobo$5bobobbob3o$9boo4bo$7bo3b3obo$8boobobbo$9bobobo$ 9bobbo$10boo5$7boo$6bobbo$5bobobo$4bobboboo$3bob3o$3bo4b3obo$4b3obo$6b o5bobo$7bobobbob3o$11boo4bo$9bo3b3obo$10boobobbo$11bobobo$11bobbo$12b oo4$7boo$6bobbo$5bobobo$4bobboboo$3bob3o3bo$3bo4boo$4b3obobbobo$6bobo$ 13bobo$8bobobbob3o$12boo4bo$10bo3b3obo$11boobobbo$12bobobo$12bobbo$13b oo$$22boo$21bobbo$20bobobo$19bobboboo$18bob3o$18bo4b3obo$16boob3obo4bo $15bobbobbo5boo$14bobobo3bobo$13bobboboo$12bob3o7bobo$12bo4b3obobbob3o $10boob3obo5boo4bo$9bobbobbo5bo3b3obo$8bobobo3bobo3boobobbo$7bobboboo 9bobobo$6bob3o7bobobbobbo$6bo4b3obobbob3oboo$4boob3obo5boo4bo$3bobbobb o5bo3b3obo$bbobobo3bobo3boobobbo$bobboboo9bobobo$ob3o7bobobbobbo$o4b3o bobbob3oboo$b3obo5boo4bo$3bo5bo3b3obo$4bobo3boobobbo$11bobobo$boo3bobo bbobbo$bobobbob3oboo$5boo4bo$3bo3b3obo$4boobobbo$5bobobo$5bobbo$6boo! It may seem that these categories encompass all possible varieties. But there are several less obvious categories. In oscillators of these types, it is impossible to join barberpoles (understood as indefinitely extended structures) to both valencies simultaneously as they will intersect each other or the body of the oscillator. But in some cases it is possible to attach other barber-active oscillators or small scraps of barberpoles. One such type, type G (though here the letter shape is a very rough approximation), has perpendicular intersecting valencies and is therefore a subtype of type L. In contrast to standard type-L radicals, the formation of homogeneous wicks from type-G radicals is not likely; it would be necessary to have enough room for the "turn". The construction of homogeneous closed paths is also generally impossible. However, it is sometimes possible to close a contour by attaching other barber-active oscillators -- type-L or quadrivalent type-X oscillators, for example -- to produce a heterogeneous closed path. #C Divalent type-G P2 radical, wick and heterogeneous oscillators x = 128, y = 62, rule = S23/B3 47bo9bo11bo9bo11bo9bo11bo9bo$47bobobobobobo11bobobobobobo11bobobobobob o11bobobobobobo$45bo6bo6bo7bo6bo6bo7bo6bo6bo7bo6bo6bo$50bobobo17bobobo 17bobobo17bobobo$44boobbo3bo3bobboo5boobbo3bo3bobboo5boobbo3bo3bobboo 5boobbo3bo3bobboo$48bo7bo13bo7bo13bo7bo13bo7bo$4boo9boo26bo3bo9bo3bo3b o3bo9bo3bo3bo3bo9bo3bo3bo3bo9bo3bo$bbobbo9bobbo24bo17bo3bo17bo3bo17bo 3bo17bo$oobbo11bobboo14boo6bobobo9bobobo3bobobo9bobobo3bobobo9bobobo3b obobo9bobobo$33bobbo12bo5bo15bo5bo15bo5bo15bo7boo$bbobboo7boobbo12boo bbo11bobbooboobbo11bobbooboobbo11bobbooboobbo11bobboo$7bobobobo$bboo6b o6boo14bobboo7boobbo5bobboo7boobbo5bobboo7boobbo5bobboo7boobbo$8bobobo 25bobobobo15bobobobo15bobobobo15bobobobo$4bobo3bo3bobo16boo6bo6boo5boo 6bo6boo5boo6bo6boo5boo6bo6boo$6bo7bo24bobobo17bobobo17bobobo17bobobo$ 35bobo3bo3bobo9bobo3bo3bobo9bobo3bo3bobo9bobo3bo3bobo$37bo7bo13bo7bo 13bo7bo13bo7bo6$105bo7bo$103bobo3bo3bobo$107bobobo$101boo6bo6boo$106bo bobobo$67bo7bo25bobboo7boobbo$65bobo3bo3bobo$69bobobo21bo3boobbo11bobb oo$63boo6bo6boo15bo5bo15bo$43bo24bobobobo16bobobbo6bobo7bobo$41bobo19b obboo7boobbo11bo$39bobobo45bo5bobo7bobo3bobo5bo$61boobbo11bobboo11bo 25bo$37bobobo21bo15bo8boo3bo3bobo7bobobo6bobbobo$65bobo7bobo14bo30bo$ 35bobo3bobo46bo8bobo3bobobo7bobo5bo$67bobo3bobo16bo28bo$33bobo7bobo43b 3o9bobobo3bobo3bobo3bo3boo$31bo15bo21bobobo18bo29bo$29boobbo11bobboo 40bo8bobobo7bobobo8bo$67bobobo20bo29bo$31bobboo7boobbo40boo3bo3bobo3bo bo3bobobo9b3o$36bobobobo22bobo3bobo19bo28bo$31boo6bo6boo41bo5bobo7bobo bo3bobo8bo$37bobobo21bobo7bobo15bo30bo$33bobo3bo3bobo15bo15bo13bobobbo 6bobobo7bobo3bo3boo$35bo7bo15boobbo11bobboo15bo25bo$95bo5bobo3bobo7bob o5bo$61bobboo7boobbo45bo$66bobobobo26bobo7bobo6bobbobo$61boo6bo6boo19b o15bo5bo$67bobobo23boobbo11bobboo3bo$63bobo3bo3bobo$65bo7bo23bobboo7b oobbo$102bobobobo$97boo6bo6boo$103bobobo$99bobo3bo3bobo$101bo7bo! Type-C oscillators have their valencies pointed towards each other, and therefore form a subtype of type I. The type-C oscillator in the example below possesses the unusual ability of self-closure: both valencies can be closed by the same segment of a barberpole. #C Divalent type-C P2 radical and closed oscillator x = 41, y = 16, rule = B3/S23 8b3o3bo17b3o3bo$14bo23bo$8boboobobbo15boboobobbo$7bo$7boo3bobboo13bobo 3bobboo$$4boo7b3o12bobo6b3o$bbobbo20bo$oobbo5bobobboo7boobbo5bobobboo$ 11bo23bo$bbobo4boo4bo10bobo4boo4bo$4bo8bo14bo8bo$bbo3bobo4bo12bo3bobo 4bo$bbo4bo4bo13bo4bo4bo$bbobo5bo15bobo5bo$4bobobobo17bobobobo! 3. Trivalent oscillators: Oscillators with barber-valency 3 are even more various. We shall discuss only their basic types, designated by the letters Y, T, F, and E, without considering the possible "intersected" connections described above for valency 2. Oscillators in which two valencies are pointed in one direction, and the third in the opposite direction, have type Y. They are reminiscent of type-I oscillators, in that they can form homogeneous wicks but cannot form closed paths. But unlike their divalent analogs, they can be combined to form agars. Generally speaking, the construction of an agar requires a 4-valent radical -- either type X or type H (these will be considered below). But if two 3-valent oscillators are connected, using up one valency each, the resulting complex will have valency 4. Thus two type-Y oscillators can be joined to form one type-H oscillator suitable for the construction of an agar. #C Trivalent type-Y P2 radical, and H-complex wick and agar x = 147, y = 67, rule = B3/S23 121bo18boo$7bo111bobobboo13bobo$5bobobboo58boo36boo7bo5bobo11bo$3bo5bo bo57bobo35bobo6bobo17bobobbo$bbobo62bo37bo10bo8bo10bo5bo$bbo8bo54bobo bbo32bobobbo32bobboo$66bo5bo31bo5bo3bobo8boo7boo8bobo$obo8boo59bobboo 33bo7bo$oobbo59boo8bobo25boo8bobo3bo5bo10bo8bo$4bo5bo108bobbobo17bobo$ 5bobbobo54bo8bo28bo8bo10bo7bo3bobo5bo$9bo62bobo35bobo6bobo7bobo7bobo$ 5bobo53bo3bobo5bo29bobo5bo15bo5bobo3bo$5boo52bobo7bobo31boobbobo7bobo 6bobo$57bo5bobo3bo37bo7bo10bo8bo$56bobo55bobobbo$56bo8bo48bo5bo3bobo8b oo$120bo7bo$54bobo8boo45boo8bobo3bo5bo$58bo70bobbobo$52bobo3bo5bo48bo 8bo10bo7bo$50bo8bobbobo55bobo6bobo7bobobboo$49bobobbo8bo45bo3bobo5bo 15bo5bobo$49bo5bo3bobo45bobo7bobo7bobo6bobo$55bo49bo5bobo3bo7bo10bo8bo $47boo8bobo44bobo17bobobbo$104bo8bo10bo5bo3bobo8boo$48bo8bo72bo7bo$55b obo44bobo8boo7boo8bobo3bo5bo$44bo3bobo5bo45boobbo32bobbobo$42bobo7bobo 51bo5bo10bo8bo10bo$40bo5bobo3bo54bobbobo17bobo6bobo$39bobo69bo7bo3bobo 5bo$39bo8bo58bobo7bobo7bobo7bobo$115bo5bobo3bo7bo$37bobo8boo55bobo6bob o17bobobbo$41bo61bo10bo8bo10bo5bo$35bobo3bo5bo54bobobbo32bobboo$33bo8b obbobo54bo5bo3bobo8boo7boo8bobo$32bobobbo8bo61bo7bo$32bo5bo3bobo55boo 8bobo3bo5bo10bo8bo$38bo78bobbobo17bobo$30boo8bobo58bo8bo10bo7bo3bobo5b o$108bobo6bobo7bobo7bobo$31bo8bo60bobo5bo15bo5bobo3bo$38bobo60boobbobo 7bobo6bobo$27bo3bobo5bo65bo7bo10bo8bo$25bobo7bobo74bobobbo$23bo5bobo3b o76bo5bo3bobo8boo$22bobo93bo7bo$22bo8bo78boo8bobo3bo5bo$127bobbobo$20b obo8boo78bo8bo10bo7bo$20boobbo93bobo6bobo7bobobboo$24bo5bo76bo3bobo5bo 15bo5bobo$25bobbobo74bobo7bobo7bobo6bobo$29bo73bo5bobo3bo7bo10bo8bo$ 25bobo74bobo17bobobbo$25boo75bo8bo10bo5bo3bobo8boo$128bo7bo$100bobo8b oo7boo8bobo3bo5bo$100boobbo32bobbobo$104bo5bo10bo8bo10bo$105bobbobo17b obo6bobo$109bo11bobo5bo7boo$105bobo13boobbobo$105boo18bo! Type T is the most interesting of the trivalent types. Radicals of this type have their valencies pointing in three out of four possible directions. Two type-T radicals can form quadrivalent complexes, including types X and H, from which it is possible to build agars and direct wicks. #C Trivalent type-T P2 radical, wick, X-complex and agars x = 242, y = 71, rule = B3/S23 193boo$191bobbo$189boobbo$$189bobobo13boo$183boo20bobbo$183bobobobo3bo bobo5boobbo$197bo$184bobbo7bobbo4bobobo13boo$185bo33bobbo$185bobobo3bo bobobobobo3bobobo5boobbo$211bo$189bobobo4bobbo7bobbo4bobobo13boo$199bo 33bobbo$189bobboo5bobobo3bobobobobobo3bobobo5boobbo$187bo3bo33bo$185b oobbo13bobobo4bobbo7bobbo4bobobo$5boo64boo6boo6boo6boo6boo6boo6boo6boo 84bo$boo3bo33boo26bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3bob obo56bobobo13bobboo5bobobo3bobobobobobo3bobobo$bobobo32bobbo26bo7bo7bo 7bo7bo7bo7bo7bo54boo20bo3bo33bo$36boobbo26bobbo4bobbo4bobbo4bobbo4bobb o4bobbo4bobbo4bobbo52bobobobo3bobobo5boobbo13bobobo4bobbo7bobbo$bobboo 56boo129bo33bo$obbo32bobobo3boo16bobobobobobobobobobobobobobobobobobob obobobobobobobobobobobobobobobo51bobbo7bobbo4bobobo13bobboo5bobobo3bob obobo$oo32bo10bo81boo52bo33bo3bo20boo$32boobbo3bobobo19bobbo4bobbo4bo bbo4bobbo4bobbo4bobbo4bobbo4bobbo57bobobo3bobobobobobo3bobobo5boobbo 13bobobo$66bo7bo7bo7bo7bo7bo7bo7bo84bo$32bobobo3bobboo17bobobo3bobobo 3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo62bobobo4bobbo7bobbo4bobobo 13bobboo$30bo11bo19boo131bo33bo3bo$28boobbo3bobobo27bobo5bobo5bobo5bob o5bobo5bobo5bobo5boo59bobboo5bobobo3bobobobobobo3bobobo5boobbo$66bo7bo 7bo7bo7bo7bo7bo7bobbo28boo27bo3bo33bo$28bobobo3bobboo23boobbo3boobbo3b oobbo3boobbo3boobbo3boobbo3boobbo3boobbo27bobbo25boobbo13bobobo4bobbo 7bobbo4bobobo$26bo11bo111boobbo54bo$24boobbo3bobobo22boo3bobobo3bobobo 3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo56bobobo13bobboo5bobobo3bobo bobobobo3bobobo$59bo65bo24bobobo20boo20bo3bo33bo$24bobobo3bobboo23bobo bo3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3boo18boo29bobobobo 3bobobo5boobbo13bobobo4bobbo7bobbo$22bo11bo109bobobobo3bobobo30bo33bo$ 20boobbo3bobobo27bobboo3bobboo3bobboo3bobboo3bobboo3bobboo3bobboo3bobb oo37bo17bobbo7bobbo4bobobo13bobboo5bobobo3bobobobo$59bobbo7bo7bo7bo7bo 7bo7bo7bo26bobbo7bobbo17bo33bo3bo20boo$20bobobo3bobboo26boo5bobo5bobo 5bobo5bobo5bobo5bobo5bobo29bo30bobobo3bobobobobobo3bobobo5boobbo13bobo bo$18bo11bo90boo23bobobo3bobobobo42bo$16boobbo3bobobo33bobobo3bobobo3b obobo3bobobo3bobobo3bobobo3bobobo3bobobo36boo20bobobo4bobbo7bobbo4bobo bo13bobboo$62bo7bo7bo7bo7bo7bo7bo7bo31bobobo36bo33bo3bo$16bobobo3bobb oo32bobbo4bobbo4bobbo4bobbo4bobbo4bobbo4bobbo4bobbo60bobboo5bobobo3bob obobobobo3bobobo5boobbo$14bo11bo29boo92bobboo24bo3bo33bo$12boobbo3bobo bo31bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo bo26bobbo24boobbo13bobobo4bobbo7bobbo4bobobo$121boo26boo54bo$12bobobo 3bobboo33bobbo4bobbo4bobbo4bobbo4bobbo4bobbo4bobbo4bobbo59bobobo13bobb oo5bobobo3bobobobobobo3bobobo$10bo11bo37bo7bo7bo7bo7bo7bo7bo7bo54boo 20bo3bo33bo$8boobbo3bobobo35bobobo3bobobo3bobobo3bobobo3bobobo3bobobo 3bobobo3bobobo54bobobobo3bobobo5boobbo13bobobo4bobbo7bobbo$56boo127bo 33bo$8bobobo3bobboo41bobo5bobo5bobo5bobo5bobo5bobo5bobo5boo52bobbo7bo bbo4bobobo13bobboo5bobobo3bobobobo$7bo10bo41bo7bo7bo7bo7bo7bo7bo7bobbo 53bo33bo3bo20boo$7boo3bobobo41boobbo3boobbo3boobbo3boobbo3boobbo3boobb o3boobbo3boobbo54bobobo3bobobobobobo3bobobo5boobbo13bobobo$199bo$12bo bboo36boo3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo58bob obo4bobbo7bobbo4bobobo13bobboo$11bobbo38bo65bo67bo33bo3bo$11boo41bobob o3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3bobobo3boo57bobboo5bobobo 3bobobobobobo3bobobo5boobbo$176bobbo33bo$54bobboo3bobboo3bobboo3bobboo 3bobboo3bobboo3bobboo3bobboo61boo13bobobo4bobbo7bobbo4bobobo$53bobbo4b obbo4bobbo4bobbo4bobbo4bobbo4bobbo4bobbo88bo$53boo6boo6boo6boo6boo6boo 6boo6boo80bobboo5bobobo3bobobobobobo3bobobo$190bobbo33bo$190boo13bobob o4bobbo7bobbo$215bo$205bobboo5bobobo3bobobobo$204bobbo20boo$204boo13bo bobo$$219bobboo$218bobbo$218boo! Type-F oscillators have three valencies, of which two are pointed in the same direction, and the third at right angles to them. These angular forms are similar to divalent type-L oscillators. Therefore the main products of polymerisation of these oscillators are zigzag wicks and self-contained homogeneous patterns. In some cases (when there is a valid U-relation) two F-radicals can form a quadrivalent complex with valencies pointing in three directions (i.e., close to type T), and from them, in turn, complexes that can form agars. However, such two-stage polymerisation is in most cases problematic. This is also true of the other mode (formation of type-H complexes, and agars from them) since at the final stage the U-relation most likely will not be valid. #C Trivalent type-F P2 radical, wicks, and oscillator x = 172, y = 50, rule = B3/S23 4boobboo57bo54bo$4bobobbo57bobo52boboboo32bo$8bo56bo54bo6bo32bobo$boo bbo57boo5boo46boo6bo31bo$bobobbobboo45bo65boo7bo24boo5boo$5boo47bobo5b o7bo46bo4bobbobobobo$bo7bo42bo5bo6boo56bo9bo21bo7bo$52bo8boobbobbobobb oo41boo9bo30boo$oo5boo34bo7bo6boo6bo5bobo47bobo8boo19boobbobbobo3bo$6b o36bobo17bo6bo43bo3bobo5bobo30bo7bobo$bbobo36bo7boobboobbo7bobo6bo37bo bo5bo3bobboo3bo23bo6bobo$4bo34boo5boo5bobo5bobo5bobo38bo5bo14bo18bo7bo bo9boo$32bo17bo6bobo7bobboobboo34bo12bobo5bo18bobobobo7bo$30bobo5bo7bo 7bo6bo47bo6boo9bobo18bo11bobobbo4bo$28bo5bo6boo7bobo5bo6boo6bo47bobo3b o18boo6bo9boo$28bo8boobbobbobo7bobobbobboo8bo34boobboobbo34boo9bo6boo$ 27bo6boo6bo5bobo7boo6bo5bo38bobo5bobo23bo4bobbobo11bo$39bo6bo7bo7bo5bo bo37bo6bobo33bo7bobobobo$25boobboobbo7bobo6bo17bo43bo6bo24boo9bobo7bo$ 29bobo5bobo5bobo5boo5boo46bobo5bo33bobo6bo$26bo6bobo7bobboobboo7bo52bo bobbobboo24bobo7bo$30bo6bo17bobo48bobo7boo30bo3bobobbobboo$26bobo5bo6b oo6bo7bo42bo11bo7bo35boo$26boobbobobbobboo8bo51bobobobo45bo7bo$34boo6b o5bo49bo12boo5boo$30bo7bo5bobo49boo6bo12bo33boo5boo$44bo55boo7bo3bobo 41bo$29boo5boo57bo4bobbobobobo5bo37bobo$35bo65bo9bo43bo$31bobo60boo9bo $33bo66bobo8boo$92bo3bobo5bobo$90bobo5bo3bobboo3bo$88bo5bo14bo$88bo12b obo5bo$87bo6boo9bobo$99bobo3bo$85boobboobbo$89bobo5bobo$86bo6bobo$90bo 6bo$86bobo5bo$86boobbobobbobboo$94boo$90bo7bo$$89boo5boo$95bo$91bobo$ 93bo! Type E is the least productive type of trivalent oscillator. It is similar to type U and allows the construction of a self-contained homogeneous oscillator of two radicals -- but only if the additional condition is met that all three attached barberpoles have homochromatic valencies (i.e., their rotors must be located on cells of one color.) Construction of wicks -- especially saturated wicks in which all connections are utilised -- is impossible in the overwhelming majority of cases, due to the lack of a valid U-relation. #C Trivalent type-E P2 radical and related oscillators x = 80, y = 21, rule = B3/S23 26boo21boo22boo$6boo17bobbo19bobbo20bobbo$6bobo16bobo20bobo21bobo$23b 3oboo17b3oboo18b3oboo$3boo3bobobo9bo6bo15bo23bo$3bobobboboobo7bobboboo 16bobbob3obo14bobbob3obo$7boo4bo7boboobobbobo12boboobo18boboobo4bo$oo 3bo4b3o6b3o4bo15b3o8bobobo8b3o8boo$obobboboobo7bo4boo6bobobo5bo4b3obo bboboobo6bo4b3obo$4boobobbo7boboobobbobobboboobo4boboobo5boo4bo6boboob o$bbo6bo9bobobo6boo4bo5bobo5bo4b3o8bobo5bobobo$3boob3o19bo4b3o9bobobbo boobo13bobobboboobo$4bobo16bobobboboobo15boobobbo17boo4bo$3bobbo20boob obbo13bo6bo11boo3bo4b3o$4boo19bo6bo15boob3o12bobobboboobo$26boob3o17bo bo18boobobbo$27bobo18bobbo16bo6bo$26bobbo19boo18boob3o$27boo41bobo$69b obbo$70boo! 4. Quadrivalent radicals: The set of quadrivalent radicals is comprised of eight basic types. We have already mentioned two of them, type-X and type-H. These are the most interesting types, since agars can be formed directly from the radicals, without creating intermediate complexes from several oscillators. Many of the eight types do not lend themselves to representation by letters. Therefore we shall use a more general system of notation for the orientations of the four valencies. The number of connections pointing in each direction will be represented by a digit, and thus the type will be designated by a sequence of four numbers. The choice of initial direction is arbitrary, as is the direction of rotation -- so that the same type may have up to eight possible labels. We shall always choose the "canonical" representation, which will be the label with the smallest first digit (and, if the first digits are equal, the smallest second digit, and so forth.) With this labeling system, the types described above will look as follows: I (0101) L (0011) U (0002) Y (0102) T (0111) F (0012) E (0003) Also, of course, a monovalent oscillator will have the label (0001), and a non-barber-active oscillator will be (0000). The eight types of quadrivalent oscillators will then be as follows: (1111) X (0112) (0121) (0202) H (0022) (0103) (0013) (0004) We see that for the first four valencies the number of possible dimensional types is set by the following sequence: 1, 1, 3, 4, 8, ... Here the first 1 corresponds to valency 0 -- i.e., an inactive oscillator. It is possible, though not trivial, to derive a formula to describe the terms of this sequence: N = m (m+1) (m+2)/3 + (m+1)(m^2 + (1+p)m + 1 + pq) Here m = [n/4] - the whole part of the quotient obtained by dividing the number of valencies n by 4, p = n mod 4 - the remainder obtained by dividing n by 4, q = [p/2] - the whole part obtained by dividing p by 2. For n = 0...12, this formula produces 1, 1, 3, 4, 8, 10, 16, 20, 29, 35, 47, 56, 72... as the first terms of the sequence. But we shall continue reviewing more concrete cases of barber-active quadrivalent oscillators. Type X or (1111): Radicals of this type have the valencies pointed in all four directions. This is most convenient type for the construction of agars. It is also possible to build wicks, but only by closing some of the connections with preblocks -- i.e., by not using all of the valencies. The smallest of the quadrivalent oscillators is a quad and a skewed quad. The agars formed by them are also widely known; they were discovered by Robert Krauz in 1970. Here is another agar based on a type-X radical: #C Quadrivalent type-X P2 oscillator, wick, and agar x = 116, y = 51, rule = B3/S23 oo4boo7boo4boo37boo4boo4boo4boo4boo4boo4boo4boo4boo4boo$obobbobo7bobo bbobo37bobobbobo4bobobbobo4bobobbobo4bobobbobo4bobobbobo$$obboobbo7bo bboobbo37bobboobbo4bobboobbo4bobboobbo4bobboobbo4bobboobbo$bo4bo9bo4bo 39bo4bo6bo4bo6bo4bo6bo4bo6bo4bo$obboobbo7bobboobbo37bobboobbo4bobboobb o4bobboobbo4bobboobbo4bobboobbo$$obobbobo7bobobbobo4boo31bobobbobo4bob obbobo4bobobbobo4bobobbobo4bobobbobo$oo4boo7boo11bo31boo52boo$22boboob o39boboobo6boboobo6boboobo6boboobo$$21b3obb3o37b3obb3o4b3obb3o4b3obb3o 4b3obb3o$$22boboobo39boboobo6boboobo6boboobo6boboobo$21bo11boo25boo52b oo$21boo4bobobbobo25bobobbobo4bobobbobo4bobobbobo4bobobbobo4bobobbobo $$27bobboobbo25bobboobbo4bobboobbo4bobboobbo4bobboobbo4bobboobbo$28bo 4bo27bo4bo6bo4bo6bo4bo6bo4bo6bo4bo$27bobboobbo25bobboobbo4bobboobbo4bo bboobbo4bobboobbo4bobboobbo$$27bobobbobo4boo19bobobbobo4bobobbobo4bobo bbobo4bobobbobo4bobobbobo$27boo11bo19boo52boo$34boboobo27boboobo6boboo bo6boboobo6boboobo$$33b3obb3o25b3obb3o4b3obb3o4b3obb3o4b3obb3o$$34bob oobo27boboobo6boboobo6boboobo6boboobo$33bo6bo19boo52boo$33boo4boo19bob obbobo4bobobbobo4bobobbobo4bobobbobo4bobobbobo$$60bobboobbo4bobboobbo 4bobboobbo4bobboobbo4bobboobbo$61bo4bo6bo4bo6bo4bo6bo4bo6bo4bo$60bobb oobbo4bobboobbo4bobboobbo4bobboobbo4bobboobbo$$60bobobbobo4bobobbobo4b obobbobo4bobobbobo4bobobbobo$60boo52boo$67boboobo6boboobo6boboobo6bob oobo$$66b3obb3o4b3obb3o4b3obb3o4b3obb3o$$67boboobo6boboobo6boboobo6bob oobo$60boo52boo$60bobobbobo4bobobbobo4bobobbobo4bobobbobo4bobobbobo$$ 60bobboobbo4bobboobbo4bobboobbo4bobboobbo4bobboobbo$61bo4bo6bo4bo6bo4b o6bo4bo6bo4bo$60bobboobbo4bobboobbo4bobboobbo4bobboobbo4bobboobbo$$60b obobbobo4bobobbobo4bobobbobo4bobobbobo4bobobbobo$60boo4boo4boo4boo4boo 4boo4boo4boo4boo4boo! Types (0121) and (0112): These types are rather similar to trivalent type-T oscillators. They can form saturated wicks and agars (these last are constructed along the same lines as type-T agars -- i.e., through quadrivalent two-radical complexes). Type H (0202): Two connections of this radical are pointed in one direction, and two others in the opposite direction. This is the other basic agar-producing type. For the immediate formation of an agar, a valid U-relation is needed for both pairs of connections. However, if the U-relation is valid for only one pair of connections, then we can connect two radicals with this pair of valencies to produce a quadrivalent type-H complex. In this complex, the U-relation will most likely be valid for both pairs of connections, and it will then be possible to construct agars. Unlike type-X radicals, type-H radicals can form completely saturated wicks. #C Quadrivalent type-H P2 oscillator, wick, and agar x = 173, y = 81, rule = B3/S23 104boobo$104bo3boo$105bo$3boo25boobo73bo$3bobo24bo3boo70bo$9b3o19bo67b oo7bo$oo3bobo25bo65bo4boo$obobo3bobo21bo67bobo3bobobo10boobo$5boo4bo 13boo7bo68bobo8b3o4bo3boo$bbo7boo13bo4boo67b3o8bobo9bo$4bo21bobo3bobob o68bobobo3bobo8bo$3bo25bobo8b3o67boo11bo$5bo19b3o8bobo68bo7bobo7bo$boo 3bo24bobobo3bobo67bo11boo$3boboo29boo70bo8bobo3bobobo10boobo$33bo7bobo bo64bobo7bobo8b3o4bo3boo$35bo10boo58boo5boob3o8bobo9bo$34bo8bo64bobo 11bobobo3bobo8bo$36bo8bo66bo14boo11bo$32boo10bo66bo12bo7bobo7bo$34bobo bo7bo57boo7bo12bo11boo$42boo60bo4boo14bo8bobo3bobobo10boobo$38bobo3bob obo56bobo3bobobo11bobo7bobo8b3o4bo3boo$41bobo8b3o53bobo8b3oboo5boob3o 8bobo9bo$37b3o8bobo53b3o8bobo7bobo11bobobo3bobo8bo$43bobobo3bobo56bobo bo3bobo8bo14boo11bo$48boo65boo11bo12bo7bobo7bo$45bo7bobobo43boobo7bo7b obo7bo12bo11boo$47bo10boo41bo3boo7bo11boo14bo8bobo3bobobo$46bo8bo46bo 10bo8bobo3bobobo11bobo7bobo8b3o$48bo8bo46bo10bo9bobo8b3oboo5boob3o8bob o$44boo10bo46bo7boo8b3o8bobo7bobo11bobobo3bobo$46bobobo7bo37boo7bo7bob obo9bobobo3bobo8bo14boo4bo$54boo40bo4boo29boo11bo12bo7boo$50bobo3bobob o36bobo3bobobo9bobobo7bo7bobo7bo12bo$53bobo8b3o33bobo8b3o8boo7bo11boo 14bo$49b3o8bobo33b3o8bobo9bo10bo8bobo3bobobo11bobo$55bobobo3bobo36bobo bo3bobo8bo10bo9bobo8b3oboo5boo$60boo45boo11bo7boo8b3o8bobo7bobo$57bo7b obobo34bo7bobo7bo7bobobo9bobobo3bobo8bo$59bo10boo34bo11boo29boo11bo$ 58bo8bo37bo8bobo3bobobo9bobobo7bo7bobo7bo$60bo8bo37bobo7bobo8b3o8boo7b o11boo$56boo10bo34boo5boob3o8bobo9bo10bo8bobo3bobobo$58bobobo7bo34bobo 11bobobo3bobo8bo10bo9bobo8b3o$66boo41bo14boo11bo7boo8b3o8bobo$62bobo3b obobo35bo12bo7bobo7bo7bobobo9bobobo3bobo$65bobo8b3o22boo7bo12bo11boo 29boo4bo$61b3o8bobo26bo4boo14bo8bobo3bobobo9bobobo7bo7boo$67bobobo3bob o24bobo3bobobo11bobo7bobo8b3o8boo7bo$72boo4bo26bobo8b3oboo5boob3o8bobo 9bo10bo$69bo7boo22b3o8bobo7bobo11bobobo3bobo8bo10bo$71bo35bobobo3bobo 8bo14boo11bo7boo3bo$70bo41boo11bo12bo7bobo7bo7boboo$72bo36bo7bobo7bo 12bo11boo$68boo3bo37bo11boo14bo8bobo3bobobo$70boboo36bo8bobo3bobobo11b obo7bobo8b3o$112bo9bobo8b3oboo5boob3o8bobo$108boo3bo4b3o8bobo7bobo11bo bobo3bobo$110boboo10bobobo3bobo8bo14boo4bo$129boo11bo12bo7boo$126bo7bo bo7bo12bo$128bo11boo14bo$127bo8bobo3bobobo11bobo$129bo9bobo8b3oboo5boo $125boo3bo4b3o8bobo7bobo$127boboo10bobobo3bobo8bo$146boo11bo$143bo7bob o7bo$145bo11boo$144bo8bobo3bobobo$146bo9bobo8b3o$142boo3bo4b3o8bobo$ 144boboo10bobobo3bobo$163boo4bo$160bo7boo$162bo$161bo$163bo$159boo3bo$ 161boboo! Types (0022), (0103) and (0013): These are close to type F, which we know is similar to type L. The most typical saturated forms for these types are self-contained homogeneous patterns from four (or more) radicals and zigzag wicks. In some cases, the formation of agars is also possible via the construction of X- or H-complexes. Type (0004): All four connections of this type are pointed in the same direction. If the valencies are homochromatic, two identical mirror-image radicals can form a closed oscillator. A valid U-relation is likely to allow the formation of wicks, though completely saturated wicks are impossible unless there is some regularity in the spacing of the valencies. 5. Summary classifications: Having considered barber-active oscillators of low valencies, we can draw the conclusion that the "chemical" properties of radicals are most closely connected to the orientations of the valencies. This appears to be increasingly true for oscillators of higher valencies. Therefore we can categorize all of the types of radicals considered above into the following six classes: Inactive (0-valent): The trivial class, included for completeness. These oscillators cannot be attached to a barberpole. Unidirectional: These comprise all types of radicals having valencies pointed in a single direction -- i.e., monovalent radicals, radicals of types U, E, (0004), (0005), etc. The general formula is (000x), where x>0. As a rule, unidirectional radicals can be used to terminate barberpoles and other wicks, and also to form two-radical oscillators. Bidirectional straight-line: The valencies of these radicals are pointed in two diametrically opposed directions. These are types I, Y, H, (0103), (0104), (0203), etc. The general formula is (0x0y), where y>=x>0. Bidirectional direct radicals are used for the construction of wicks. Bidirectional angular: Here the valencies are pointed in two perpendicular directions. These are types L, F, (0013), (0022), etc. The general formula is (00xy), where y>=x>0. Bidirectional angular radicals can be used for the construction of zigzag wicks, as well as closed paths containing a minimum of 4 radicals. Three-directional: Oscillators of this class have the valencies pointing in three different directions, with only valencies of the fourth direction not present. These are oscillators of types T, (0112), (0121), etc. The general formula is (0xyz), where z>=x>0, y>0. Three-directional radicals can be used for the construction of homogeneous wicks and agars -- but only through the formation of intermediate bidirectional or pandirectional complexes. Pandirectional: These oscillators have valencies pointed in all four possible directions. They include our type-X oscillators, and also higher-valency types not discussed here such as (1112), (1113), (1122), (1212), etc. The general formula is (xyzt), where t>=y>=x>0, z>=x. Pandirectional radicals are the basic structural material for the formation of agars. Nicolay Beluchenko, April, 2004.