Chapter IX: On Concretions, Spicules, and Spicular Skeletons (3)

Let us next suppose that, upon some such lattice-work as has just been described, another and external layer of cells or vesicles is developed, and that instead of (or perhaps only in addition to) a second hexagonal lattice-work, which might develop concentrically to the first in the boundary-furrows of this new layer of cells, the siliceous matter now tends to be deposited radially, or normally to the surface of the sphere, just in the lines where the external layer of vesicles meet one another, three by three. The result will be that, when the vesicles themselves are removed, a series of radiating spicules will be revealed, directed outwards from each of the angles of the original hexagon; as is seen in Fig. 225. And it may further happen that these radiating skeletal rods are continued at their distal ends into divergent rays, forming a triple fork, and corresponding (after a fashion {472} which we have already described as occurring in certain sponge-spicules) to the three superficial furrows between the adjacent cells. This last is, as it were, an intermediate stage between the simple rods and the complete formation of another concentric sphere of latticed hexagons. Another possible case is when the large and uniform vesicles of the outer protoplasm are mixed with, or replaced by, much smaller vesicles, piled on one another in more or less concentric layers; in this case the radiating rods will no longer be straight, but will be bent into a zig-zag pattern, with angles in three vertical planes, corresponding to the successive contacts of the groups of cells around the axis (Fig. 226).

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Among a certain group called the Nassellaria, we find geometrical forms of peculiar simplicity and beauty,—such for instance as that which I have represented in Fig. 227. It is obvious at a glance that this is such a skeleton as may have been formed {473} (I think we may go so far as to say _must_ have been formed) at the interfaces of a little tetrahedral group of cells, the four equal cells of the tetrahedron being in this particular case supplemented by a little one in the centre of the system. We see, precisely as in the internal boundary-system of an artificial group of four soap-bubbles, the plane surfaces of contact, six in number; the relation to one another of each triple set of interfacial planes, meeting one another at equal angles of 120°; and finally the relation of the four lines or edges of triple contact, which tend (but for the little central vesicle) to meet at co-equal solid angles in the centre of the system, all as we have described on p. 318. In short, each triple-walled re-entrant angle of the little shell has essentially the configuration (or a part thereof) of what we have called a “Maraldi pyramid” in our account of the architecture of the honeycomb, on p. 329[484].

There are still two or three remarkable or peculiar features in this all but mathematically perfect shell, and they are in part easy and in part they seem more difficult of interpretation.

We notice that the amount of solid matter deposited in the plane interfacial boundaries is greatly increased at the outer margin of each boundary wall, where it merges or coincides with the superficial furrow which separates the free, spherical surfaces of the bubbles from one another; and we may sometimes find that, along these edges, the skeleton remains complete and strong, while it shows signs of imperfect development or of breaking away over great part of the rest of the interfacial surfaces. In this there is nothing anomalous, for we have already recognised that it is at the edges or margins of the interfacial partition-walls that the manifestation of surface-energy will tend to reach its maximum. And just as we have seen that, in certain of our “multicellular” spherical Radiolarians, it is at the superficial {474} edges or borders of the partitions, and here only, that skeletal formation occurs (giving rise to the netted shell with its hexagonal meshes of Fig. 221), so also at times, in the case of such little aggregates of cells or vesicles as the four-celled system of Callimitra, it may happen that about the external boundary-_lines_, and not in the interior boundary-_planes_, the whole of the skeletal matter is aggregated. In Fig. 228 we see a curious little skeletal structure or complex spicule, whose conformation is easily accounted for after this fashion. Little spicules such as this form isolated portions of the skeleton in the genus _Dictyocha_, and occur scattered over the spherical surface of the organism (Fig. 229). The more or less basket-shaped spicule has evidently been developed about a little cluster of four cells or vesicles, lying in or on the plane of the surface of the organism, and therefore arranged, not in the tetrahedral form of Callimitra, but in the manner in which four contiguous cells lying side by side normally set themselves, like the four cells of a segmenting egg: that is to say with an intervening “polar furrow,” whose ends mark the meeting place, at equal angles, of the cells in groups of three.

The little projecting spokes, or spikes, which are set normally to the main basket-work, seem to be incompleted portions of a larger basket, or in other words imperfectly formed elements corresponding to the interfacial contacts in the surrounding parts {475} of the system. Similar but more complex formations, all explicable as basket-like frameworks developed around a cluster of cells, are known in great variety.

In our Nassellarian itself, and in many other cases where the plane interfacial boundary-walls are skeletonised, we see that the siliceous matter is not deposited in an even and continuous layer, like the waxen walls of a bee’s cell, but constitutes a meshwork of fine curvilinear threads; and the curves seem to run, on the whole, isogonally, and to form three main series, one approximately parallel to, or concentric with, the outer or free edge of the partition, and the other two related severally to its two edges of attachment. Sometimes (as may also be seen in our figure), the system is still further complicated by a fourth series of linear elements, which tend to run radially from the centre of the system to the free edge of each partition. As regards the former, their arrangement is such as would result if deposition or solidification had proceeded in waves, starting independently from each of the three boundaries of the little partition-wall; and something of this kind is doubtless what has happened. We are reminded at once of the wave-like periodicity of the Liesegang phenomenon. But apart from this we might conceive of other explanations. For instance, the liquid film which originally constitutes the partition must easily be thrown into _vibrations_, and (like the dust upon a Chladni’s plate) minute particles of matter in contact with the film would tend to take up their position in a symmetrical arrangement, in direct relation to the nodal points or lines of the vibrating surface[485]. Some such explanation as this (to my thinking) must be invoked to account for the minute and varied and very beautiful patterns upon many diatoms, the resemblance of which patterns (in certain of their simpler cases) to the Chladni figures is sometimes striking and obvious. But the many special problems which the diatom skeleton suggests I have not attempted to consider.

The last peculiarity of our Nassellarian lies in an apparent departure from what we should at first expect in the way of its {476} external symmetry. Were the system actually composed of four spherical vesicles in mutual contact, the outer margin of each of the six interfacial planes would obviously be a circular arc; and accordingly, at each angle of the tetrahedron, we should expect to have a depressed, or re-entrant angle, instead of a prominent cusp. This is all doubtless due to some simple balance of tensions, whose precise nature and distribution is meanwhile a matter of conjecture. But it seems as though an extremely simple explanation would go a long way, and possibly the whole way, to meet this particular case. In our ordinary plane diagram of three cells, or soap-bubbles, in contact, we know (and we have just said) that the tensions of the three partitions draw inwards the outer walls of the system, till at each point of triple contact (_P_) we tend to get a triradiate, equiangular junction. But if we introduce another bubble into the centre of the system (Fig. 230), then, as Plateau shewed, the tensions of its walls and those of the three partitions by which it is now suspended, again balance one another, and the central bubble appears (in plane projection) as a curvilinear, equilateral triangle. We have only got to convert this plane diagram into that of a tetrahedral solid to obtain _almost_ precisely the configuration which we are seeking to explain. Now we observe that, so far as our figure of Callimitra informs us, this is just the shape of the little bubble which occupies the centre of the tetrahedral system in that Radiolarian skeleton. And I conceive, accordingly, that the entire organism was not limited to the four cells or vesicles (together with the little central {477} fifth) which we have hitherto been imagining, but there must have been an outer tetrahedral system, enclosing the cells which fabricated the skeleton, just as these latter enclosed, and deformed, the little bubble in the centre of all. We have only to suppose that this hypothetical tetrahedral series, forming the outer layer or surface of the whole system, was for some chemico-physical reason incapable of secreting at its interfacial contacts a skeletal fabric[486].

In this hypothetical case, the edges of the skeletal system would be circular arcs, meeting one another at an angle of 120°, or, in the solid pyramid, of 109°: and this latter is _very nearly_ the condition which our little skeleton actually displays. But we observe in Fig. 227 that, in the immediate neighbourhood of the tetrahedral angle, the circular arcs are slightly drawn out into projecting cusps (cf. Fig. 230, _B_). There is no S-shaped curvature of the tetrahedral edges as a whole, but a very slight one, a very slight change of curvature; close to the apex. This, I conceive, is nothing more than what, in a material system, we are bound to have, to represent a “surface of continuity.” It is a phenomenon precisely analogous to Plateau’s “bourrelet,” which we have already seen to be a constant feature of all cellular systems, rounding off the sharp angular contacts by which (in our more elementary treatment) we expect one film to make its junction with another[487].

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In the foregoing examples of Radiolaria, the symmetry which the organism displays would seem to be identical with that symmetry of forces which is due to the assemblage of surface-tensions in the whole system; this symmetry being displayed, in one class of cases, in a complex spherical mass of froth, and in {478} another class in a simpler aggregate of a few, otherwise isolated, vesicles. But among the vast number of other known Radiolaria, there are certain forms (especially among the Phaeodaria and Acantharia) which display a still more remarkable symmetry, the origin of which is by no means clear, though surface-tension doubtless plays a part in its causation. These are cases in which (as in some of those already described) the skeleton consists (1) of radiating spicular rods, definite in number and position, and (2) of interconnecting rods or plates, tangential to the more or less spherical body of the organism, whose form becomes, accordingly, that of a geometric, polyhedral solid. It may be that there is no mathematical difference, save one of degree, between such a hexagonal polyhedron as we have seen in _Aulacantha_, and those which we are about to describe; but the greater regularity, the numerical symmetry, and the apparent simplicity of these latter, makes of them a class apart, and suggests problems which have not been solved nor even investigated.

The matter is sufficiently illustrated by the accompanying figures, all drawn from Haeckel’s Monograph of the Challenger Radiolaria[488]. In one of these we see a regular octahedron, in another a regular, or pentagonal dodecahedron, in a third a regular icosahedron. In all cases the figure appears to be perfectly symmetrical, though neither the triangular facets of the octahedron and icosahedron, nor the pentagonal facets of the dodecahedron, are necessarily plane surfaces. In all of these cases, the radial spicules correspond to the solid angles of the figure; and they are, accordingly, six in number in the octahedron, twenty in the dodecahedron, and twelve in the icosahedron. If we add to these three figures the regular tetrahedron, which we have had frequent occasion to study, and the cube (which is represented, at least in outline, in the skeleton of the hexactinellid sponges), we have completed the series of the five regular polyhedra known to geometers, the _Platonic bodies_[489] of the older mathematicians. It is at first sight all the more remarkable that we should here meet {480} with the whole five regular polyhedra, when we remember that, among the vast variety of crystalline forms known among minerals, the regular dodecahedron and icosahedron, simple as they are from the mathematical point of view, never occur. Not only do these latter never occur in Crystallography, but (as is explained in text-books of that science) it has been shewn that they cannot occur, owing to the fact that their indices (or numbers expressing the relation of the faces to the three primary axes) involve an irrational quantity: whereas it is a fundamental law of crystallography, involved in the whole theory of space-partitioning, that “the indices of any and every face of a crystal are small whole numbers[490].” At the same time, an imperfect pentagonal dodecahedron, whose pentagonal sides are non-equilateral, is common among crystals. If we may safely judge from Haeckel’s figures, the pentagonal dodecahedron of the Radiolarian is perfectly regular, and we must presume, accordingly, that it is not brought about by principles of space-partitioning similar to those which manifest themselves in the phenomenon of crystallisation. It will be observed that in all these radiolarian polyhedral shells, the surface of each external facet is formed of a minute hexagonal network, whose probable origin, in relation to a vesicular structure, is such as we have already discussed.

In certain allied Radiolaria (Fig. 232), which, like the dodecahedral form figured in Fig. 231, 5, have twenty radial spines, these latter are commonly described as being arranged in a certain very singular way. It is stated that their arrangement may be referred {481} to a series of five parallel circles on the sphere, corresponding to the equator (_c_), the tropics (_b_, _d_) and the polar circles (_a_, _e_); and that beginning with four equidistant spines in the equator, we have alternating whorls of four, radiating outwards from the sphere in each of the other parallel zones. This rule was laid down by the celebrated Johannes Müller, and has ever since been used and quoted as Müller’s law. The chief point in this alleged arrangement which strikes us at first sight as very curious, is that there is said to be no spine at either pole; and when we come to examine carefully the figure of the organism, we find that the received description does not do justice to the facts. We see, in the first place, from such figures as Figs. 232, 234, that here, unlike our former cases, the radial spines issue through the facets (and through _all_ the facets) of the polyhedron, instead of through its solid angles; and accordingly, that our twenty spines correspond (not, as before, to a dodecahedron) but to some sort of an icosahedron. We see in the next place, that this icosahedron is composed of faces, or plates, of two different kinds, some hexagonal and some pentagonal; and when we look closer, we discover that the whole figure is that of a hexagonal prism, whose twelve solid angles are replaced by pentagonal facets. Both hexagons and pentagons {482} appear to be perfectly equilateral, but if we try to construct a plane-sided polyhedron of this kind, we soon find that it is impossible; for into the angles between the six equatorial hexagons those of the six united pentagons will not fit. The figure however can be easily constructed if we replace the straight edges (or some of them) by curves, and the plane facets by corresponding, slightly curved, surfaces. The true symmetry of this figure, then, is hexagonal, with a polar axis, produced into two polar spicules; with six equatorial spicules, or rays; and with two sets of six spicular rays, interposed between the polar axis and the equatorial rays, and alternating in position with the latter.

Müller’s description was emended by Brandt, and what is now known as “Brandt’s law,” viz. that the symmetry consists of two polar rays, and three whorls of six each, coincides with the above description so far as the spicular axes go: save only that Brandt specifically states that the intermediate whorls stand equidistant between the equator and the poles, i.e. in latitude 45°. While not far from the truth, this statement is not exact; for according to the geometry of the figure, the intermediate cycles obviously stand in a slightly higher latitude, but this latitude I have not attempted to determine; for the calculation seems to be a little troublesome owing to the curvature of the sides of the figure, and the enquiring mathematician will perform it more easily than I. Brandt, if I understand him rightly, did not propose his “law” as a substitute for Müller’s law, but as a second law applicable to a few particular cases. I on the other hand can find no case to which Müller’s law properly applies.

If we construct such a polyhedron, and set it in the position of Fig. 232, _B_, we shall easily see that it is capable of explanation (though improperly) in accordance with Müller’s law; for the four equatorial rays of Müller (_c_) now correspond to the two polar and to two opposite equatorial facets of our polyhedron: the four “polar” rays of Müller (_a_ or _e_) correspond to two adjacent hexagons and two intermediate pentagons of the figure: and Müller’s “tropical” rays (_b_ or _d_) are those which emanate from the remaining four pentagonal facets, in each half of the figure. In some cases, such as Haeckel’s _Phatnaspis cristata_ (Fig. 233), we have an ellipsoidal body, from which the spines emerge in the order described, but which is not obviously divided by facets. In Fig. 234 I have indicated the facets corresponding to the rays, and dividing the surface in the usual symmetrical way. {483}

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Within any polyhedron we may always inscribe another polyhedron, whose corners correspond in number to the sides or facets of the original figure, or (in alternative cases) to a certain number of these sides; and a similar result is obtained by bevelling off the corners of the original polyhedron. We may obtain a precisely similar symmetrical result if (in such a case as these Radiolarians which we are describing), we imagine the radial spines to be interconnected by tangential rods, instead of by the complete facets which we have just been dealing with. In our complicated polyhedron with its twenty radial spines arranged in the manner described there are various symmetrical ways in which we may imagine these interconnecting bars to be arranged. The most symmetrical of these is one in which the whole surface is divided into eighteen rhomboidal areas, obtained by systematically connecting each group of four adjacent radii. This figure has eighteen faces (_F_), twenty corners (_C_), and therefore thirty-six edges (_E_), in conformity with Euler’s theorem, _F_ + _C_ = _E_ + 2.

Another symmetrical arrangement will divide the surface into fourteen rhombs and eight triangles. This latter arrangement is obtained by linking up the radial rods as follows: _aaaa_, _aba_, _abcb_, _bcdc_, etc. Here we have again twenty corners, but we have twenty-two faces; the number of edges, or tangential spicular bars, will be found, therefore, by the above formula, to be forty. In Haeckel’s figure of _Phractaspis prototypus_ we have a spicular skeleton which appears to be constructed precisely upon this plan, and to be derivable from the faceted polyhedron precisely after this manner.

In all these latter cases it is the arrangement of the axial rods, or in other words the “polar symmetry” of the entire organism, which lies at the root of the matter, and which, if only {485} we could account for it, would make it comparatively easy to explain the superficial configuration. But there are no obvious mechanical forces by which we can so explain this peculiar polarity. This at least is evident, that it arises in the central mass of protoplasm, which is the essential living portion of the organism as distinguished from that frothy peripheral mass whose structure has helped us to explain so many phenomena of the superficial or external skeleton. To say that the arrangement depends upon a specific polarisation of the cell is merely to refer the problem to other terms, and to set it aside for future solution. But it is possible that we may learn something about the lines in which _to seek for_ such a solution by considering the case of Lehmann’s “fluid crystals,” and the light which they throw upon the phenomena of molecular aggregation.

The phenomenon of “fluid crystallisation” is found in a number of chemical bodies; it is exhibited at a specific temperature for each substance; and it would seem to be limited to bodies in which there is a more or less elongated, or “chain-like” arrangement of the atoms in the molecule. Such bodies, at the appropriate temperature, tend to aggregate themselves into masses, which are sometimes spherical drops or globules (the so-called “spherulites”), and sometimes have the definite form of needle-like or prismatic crystals. In either case they remain liquid, and are also doubly refractive, polarising light in brilliant colours. Together with them are formed ordinary solid crystals, also with characteristic polarisation, and into such solid crystals all the fluid material ultimately turns. It is evident that in these liquid crystals, though the molecules are freely mobile, just as are those of water, they are yet subject to, or endowed with, a “directive force,” a force which confers upon them a definite configuration or “polarity,” the _Gestaltungskraft_ of Lehmann.

Such an hypothesis as this had been gradually extruded from the theories of mathematical crystallography[491]; and it had come to be believed that the symmetrical conformation of a homogeneous crystalline structure was sufficiently explained by the mere mechanical fitting together of appropriate structural units along the easiest and simplest lines of “close packing”: just as {486} a pile of oranges becomes definite, both in outward form and inward structural arrangement, without the play of any _specific_ directive force. But while our conceptions of the tactical arrangement of crystalline molecules remain the same as before, and our hypotheses of “modes of packing” or of “space-lattices” remain as useful as ever for the definition and explanation of the molecular arrangements, an entirely new theoretical conception is introduced when we find such space-lattices maintained in what has hitherto been considered the molecular freedom of a liquid field; and we are constrained, accordingly, to postulate a specific molecular force, or “Gestaltungskraft” (not unlike Kepler’s “facultas formatrix”), to account for the phenomenon.

Now just as some sort of specific “Gestaltungskraft” had been of old the _deus ex machina_ accounting for all crystalline phenomena (_gnara totius geometriæ, et in ea exercita_, as Kepler said), and as such an hypothesis, after being dethroned and repudiated, has now fought its way back and has made good its right to be heard, so it may be also in biology. We begin by an easy and general assumption of _specific properties_, by which each organism assumes its own specific form; we learn later (as it is the purpose of this book to shew) that throughout the whole range of organic morphology there are innumerable phenomena of form which are not peculiar to living things, but which are more or less simple manifestations of ordinary physical law. But every now and then we come to certain deep-seated signs of protoplasmic symmetry or polarisation, which seem to lie beyond the reach of the ordinary physical forces. It by no means follows that the forces in question are not essentially physical forces, more obscure and less familiar to us than the rest; and this would seem to be the crucial lesson for us to draw from Lehmann’s surprising and most beautiful discovery. For Lehmann seems actually to have demonstrated, in non-living, chemical bodies, the existence of just such a determinant, just such a “Gestaltungskraft,” as would be of infinite help to us if we might postulate it for the explanation (for instance) of our Radiolarian’s axial symmetry. But further than this we cannot go; for such analogy as we seem to see in the Lehmann phenomenon soon evades us, and refuses to be pressed home. Not only is it the case, as we have already {487} seen, that certain of the geometric forms assumed by the symmetrical Radiolarian shells are just such as the “space-lattice” theory would seem to be inapplicable to, but it is in other ways obvious that symmetry of _crystallisation_, whether liquid or solid, has no close parallel, but only a series of analogies, in the protoplasmic symmetry of the living cell.

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