(Part 3 of a series of articles on oscillator "chemistry" in the Game of Life.)

A "radical" is a periodic object that maintains its structure when other objects interact with it at one or several of its several boundary sites.

These boundary sites are the "valencies" or "linkages" of a radical.

Radicals can routinely be found as parts of larger periodic Life objects, such as oscillators, wicks, agars, spaceships, waves, and fuses. Typical variants of radicals are fenceposts, induction coils, tagalongs, components of spaceships, etc.

Here are the basic features of radicals and valencies:

1. The period: The period of a radical is the minimum time (number of generations) necessary for a given configuration to recur. The period of any linkage in a radical is always a factor of the period of the radical. In the wick shown below, the period of the "barber-valency" linkages is 2, whereas the period of the radicals that form the wick is 4.

Even more common are examples when radicals of higher periods are connected to valencies of period 1 -- by an inductive or tie linkage, for example. However, such cases are routinely considered to be trivial linkages. The most interesting variants are the ones where the periods of the radical and valency coincide. Up to now we have only considered cases of period 2, the simplest oscillator period.

It is simple enough to construct a chemistry for valencies of period 1. I shall not do it in full detail here; I shall note only that it is possible to define two types of valencies as the basic varieties: inductive linkage, where radicals do not touch each other, and immediate linkage, where the radicals are in direct contact. These categories of valencies exist for other periods, but they are particularly important for period 1. For other periods inductive linkage is much less common, except for the special case where mirror-image halves of an oscillator maintain an inductive linkage across a row of empty cells: birth in the central row is impossible because the number of neighbors is always even.

Normally the period of a radical coincides with, or at least is a factor of, the period of the object of which the radical is a part. However, it will be shown below in the 'ants' example below that this is not always the case! An infinite ant trail may be considered to move orthogonally at one-fifth or two-fifths of the speed of light, or various intermediate speeds depending on the offsets of the individual ants in the trail. But a finite segment of ant trail with one-cell offsets can travel diagonally at c/4 instead, if it is attached to a spaceship traveling diagonally at one fourth of the speed of light that creates and destroys ant-radicals at the appropriate connection points.

2. Displacement and velocity: The displacement of a radical (and therefore of its linkages) is the distance that the radical moves during the period. The velocity is then the displacement of a radical divided by its period. The introduction of these concepts of displacement and velocity of a radical allows us to extend the nomenclature of oscichemistry to such moving objects as spaceships and waves. Probably there is no particular need for such an extension, as the structure of spaceships is adequately described within the framework of the existing concept of spaceship "grammar". But the new nomenclature can be shown to be equivalent, so it can perhaps provide a fresh look at known truths.

Radicals are referred to as 'components' of a grammar, but there is no specific term in the nomenclature of grammars for valency or for linkage. It is possible to note only that in tabulated grammars each type of linkage corresponds to a line in the table. The components in the table to the left of the vertical line have valencies of one polarity (let's say positive), and the components to the right then have the other (negative) polarity. If the same component is located on both sides the of vertical line (probably in different rows) it is a diradical.

Radicals of higher valencies, as well as radicals with covalent linkages in grammars, have not been investigated in much detail, probably because they play a smaller role in structure formation of spaceships.

The velocity of a spaceship's radicals (component) routinely coincides with the velocity of the spaceship.

However, in Life chemistry situations are possible where certain parts of an object, which we consider to be its radicals, move with velocities and in directions which are distinct from the velocity and direction of movement of the object. A typical example is a wick made up of eight-bit "ants". Each ant is a diradical, with two linkages of various polarities connected with the two adjacent ants. The period of such a radical is 1, and the displacement is also 1, so the travelling speed of an ant is equal to the speed of light.

We shall now limit an "ants" wick by two fenceposts, thus converting it to an oscillator. The velocity and direction of movement of the ants in this oscillator will not vary, but the travelling speed of the oscillator as whole is zero. The period (5) also differs from the period of each separate ant. Other mobile terminations of ants can convert them into a period-4 spaceship with velocity 1/4, or a period-6 spaceship with velocity 1/6 (for period 6, no concrete examples have yet been found).

Also possible (though more difficult to search for) would be spaceships of other velocities that contain these same ants. The simplest variant is a period-4 spaceship moving diagonally with velocity 1/4 that includes a line of ants with a lateral displacement of 1 step. In this case the ants and the spaceship differ not only in their velocities, but also in their directions of motion: the spaceship moves diagonally with velocity 1/4, while individual ants move orthogonally with a velocity of 1.

In these cases, the terminal elements are interesting objects. These appear to be radicals that share the velocity and period of the ship (or oscillator). But the terminal elements' linkages to the first and last ants in an 'ant trail' can be interpreted in two ways: the linkage can be seen as having either the velocity and period of the termination, or the velocity and period of an ant. In the second case the valency is transient: it arises together with an ant at one terminal element, moves together with it and is destroyed when the ant is absorbed by the other terminal element.

3. Type of valency: This is defined by structure of a radical and the reciprocal radical connected to it. Radicals of one type can form homogeneous complexes, since they can interact stably with copies of themselves. Radicals of the other type cannot be connected immediately with each other.

4. Polarity of valency: In standard chemistry, valencies can be polar or covalent. In the chemistry of oscillators, there is also a third transition type, where the polarity is reversed halfway through the valency's period. It is convenient to refer to these as "semipolar" valencies.

Radicals with covalent linkages can be connected directly to duplicates of themselves, as well as to other radicals with the same type of valency.

Polar linkages have a directedness, and connections can only be made between radicals with linkages of different polarities. In particular, radicals having polar linkages cannot be connected to duplicates of themselves.

Donor-acceptor linkages are a subtype of polar linkage where the polarization is expressed most clearly. In this case one radical (the donor) makes a spark necessary for maintaining the other radical's stability. In this case it is expedient always to assign positive polarity to the donor, and negative polarity to the acceptor. A common use of donor-acceptor linkages is in the stabilization of oscillators with sparkers of various types.

We considered semipolar linkages by the example of period-2 dot linkage. These linkages reverse their polarity every half cycle, and thus such linkages must have even periods. Semipolar linkages can form a smaller number of dimensional types of multivalent radicals, as compared to polar linkages. Theoretically semipolar donor-acceptor linkages might exist that incorporate radicals which in turn provide the sparks needed by other radicals; however, I do not know an oscillator or a spaceship containing such a linkage.

5. Symmetry of linkage: In discussing linkage symmetry, we will be concerned only with symmetry relative to an axis parallel to the linkage direction. [Generally speaking, polarity -- or rather its lack, i.e., covalence -- is also a kind of symmetry, but with an axis perpendicular to the linkage direction.]

Linkage symmetry influences the number of possible dimensional types of multivalent radicals, and, as a corollary, the varieties of objects that can be produced with the help of this or that radical. Two types of symmetry are possible. The first is simple mirror symmetry, where the axis of symmetry either passes through the centers of a line of cells, or (for axes parallel to the X or Y axes) down the middle of a doubled row or column of cells. The second type of symmetry involves a glide reflection, where a reflected copy must be shifted along the axis of a symmetry in order to match the original. The latter type of symmetry is characteristic of barber-valencies, for example. For barber-valencies the axis of symmetry passes through the centers of a diagonal line of cells; glide-reflective symmetry also allows diagonal axes that fall between cell centers.

6. Number and orientation of valencies: This parameter determines the possible range of application of a radical. For example, the only homogeneous oscillators that can be formed by monovalent radicals with non-polar linkages are those consisting of two radicals; monovalent radicals with polar linkages cannot form homogeneous objects at all. In inhomogeneous objects, monovalent radicals serve as final elements of larger oscillators or as fenceposts for wicks.

Diradicals are a structural material for homogeneous wicks, and (depending on the valency orientation) for oscillators containing a closed path.

Radicals of higher valencies, especially those with valencies pointing in all four directions, can be applied to building agars. Previous articles in this series have discussed this topic in detail.

The first article ("Barber-chemistry") enumerated the classes of multivalent radicals for the elementary case of symmetric covalent linkages. It appears that the corresponding formulas for linkages of other types are much more complicated. Here I shall summarize only the number of dimensional types for monovalent radicals and diradicals.

Number of dimensional types of 1-valent radicals

Number of dimensional types of 2-valent radicals

Extending the analogy between oscichemistry and spaceship grammars, it is interesting to note the existence of fixed radicals similar to tagalongs. Such radicals can be removed from the basic oscillator, and last will keep the stability. It is logical to name such radicals 'oscillongs', or 'stillongs' for period 1. For example, the following two small oscillongs to the caterer are widely known.


Examples of stillongs are even simpler: in the following still-lifes the stillongs are the complex objects to the right of the block and tub.


The ability of a radical to be an oscillong is not an intrinsic property of the radical -- it depends also on the oscillator to which the radical is connected. (The equivalent is true of components: to define a tagalong it is necessary to specify a connection point on a particular spaceship.) Thus, in first of the examples given below, the long bookend is a stillong to the block; in the second, both long bookends are only radicals, not stillongs at all.


Oscillongs are also routinely found where the oscillator engines have been stabilized with the help of sparkers; removal of the engine does not normally influence the stability of sparkers (though, certainly, this deprives a beautiful oscillator of any sense). In the stable case, a Herschel moving in a loop also can be an oscillong to the conduit. This last is strictly true if the conduit contains eaters, blocks or other still-lifes, unless they are all functioning as 'rocks' (are there such conduits?) Usually a conduit is much poorer without Herschels than with them; for example, the eaters in an empty conduit are merely still-lifes, whereas a Herschel makes them 'open their mouths' with the necessary period (that is, in the presence of a Herschel they become radicals).


Experience shows that the use of oscichemistry frequently helps to find new objects. Searching for radicals with the necessary period and displacement is a much easier problem than searching for complete objects, largely because such radicals tend to have significantly smaller sizes. After detecting a great many mono- and multivalent radicals with different types of linkages, sooner or later it is possible to build complete objects from them.

A similar approach is already commonly applied in designing new spaceships. Oscillator searches are routinely done by other methods; but studying the "chemical" nature of oscillators has allowed me to discover and construct many new objects while writing this series of articles -- even in such a heavily-studied area as period-2 oscillators.

Nicolay Beluchenko, April, 2004, with provisional editing by Dave Greene, July 2009