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.

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.

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.

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 would intersect each other -- or, in some cases, 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. Type G is therefore a sub-type of type L -- but homogeneous wicks using type-G radicals are not usually possible; 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.

Type-C oscillators have their valencies pointed towards each other, and thus type C is a subset 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.

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.

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.

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) it is possible to form of two F-radicals the quadrivalent complex having tripartite orientation of valencies (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.

Type E is the least productive type of trivalent oscillator. It is similar to type U and may allow the construction of a self-contained homogeneous oscillator from two mirror-image radicals. However, in order for this to work, the radical must have homochromatic valencies: all three attached barberpoles must have their rotors located on cells of one color.

It is also possible to construct a saturated oscillator from two identical radicals oriented 180 degrees from each other. In this case, the distances between the valencies must be uniform, and the phases of the valencies must be coordinated as well.

Construction of wicks from type-E radicals -- especially saturated wicks (i.e., wicks in which all connections are utilised) -- is impossible in the overwhelming majority of cases, due to the lack of a valid U-relation.

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 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 figure, and so forth.)

With this labeling system, the types described above will look as follows:

(0101) I

(0011) L

(0002) U

(0102) Y

(0111) T

(0012) F

(0003) E

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 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 are the quad and skewed quad. The agars formed by these are also widely known; they were discovered by Robert Krauz in 1970. Here is another agar based on a type-X radical:

**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.

**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. A saturated oscillator composed of two identical mirror-image radicals is possible. 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.

To connect two 180-degree rotated copies of a (0004) radical -- or two such copies of any (000n) radical, for that matter -- the spacing of the valencies must be palindromic. (This condition enforces uniform spacing in the case of E-type radicals, above.) The phases of connected valencies must also be coordinated.

**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 the basic unit of repetition in wicks, as well as in one of the two basic types of agars.

**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 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 other basic structural material for the formation of agars.

Nicolay Beluchenko, April, 2004.