On Growth and Form

* Apart from the fact that the apex of each pyramid is interrupted, or truncated, by the presence of the little central cell, it is also possible that the solid angles are not precisely equivalent to those of Maraldi’s pyramids, owing to the fact that there is a certain amount of distortion, or axial asymmetry, in the Nassellarian system. In other words (to judge from Haeckel’s figures), the tetrahedral symmetry in Nassellaria is not absolutely regular, but has a main axis about which three of the trihedral pyramids are symmetrical, the fourth having its solid angle somewhat diminished.

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 boundaryplanes, 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 ofthesystem. 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 wibrations, 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 *. 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 hes in an apparent departure from what we should at first expect in the way of its 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

fifth) which we have hitherto been imagining, but there must have been an outer tetrahedral system, enclosing the cells which fabri- - cated 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 *.

In this hypothetical case, the edges of the skeletal system would be circular ares, 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 ares 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 f.

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 surfacetensions in the whole system; this symmetry being displayed, in one class of cases, in a complex spherical mass of froth, and in * We need not go so far as to suppose that the external layer of cells wholly lacked the power of secreting a skeleton. In many of the Nassellariae figured by Haeckel (for there are many variant forms or species besides that represented here), the skeleton of the partition-walls is very slightly and scantily developed. In such a case, if we imagine its few and scanty strands to be broken away, the central tetrahedral figure would be set free, and would have all the appearance of a complete and independent structure.

+ The “bourrelet” is not only, as Plateau expresses it, a “surface of continuity,” but we also recognise that it tends (so far as material is available for its production) to further lessen the free surface-area. On its relation to vapour-pressure and to the stability of foam, see: FitzGerald’s interesting note in Nature, Feb. 1, 1894 (Works, p. 309). 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 sohd. 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*. 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} of the older mathematicians. It is at first sight all the more remarkable that we should here meet

* Of the many thousand figures in the hundred and forty plates of this beautifully illustrated book, there is scarcely one which does not depict, now patently, now in pregnant suggestion, some subtle and elegant geometrical configuration. + They were known (of course) long before Plato: [Adrwy dé Kal év rovrous mubaryopifer. Skeletons of various Radiolarians, after Haeckel. furcus; 2. C. octahedrus; 3. Circogonia icosahedra; 4. Circospathi, 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*.” 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 * Tf the equation of any plane face of a crystal be written in the form ha +ky+lz=1, then h, k, l are the indices of which we are speaking. They are the reciprocals of the parameters, or reciprocals of the distances from the origin at which the plane meets the several axes. In the case of the regular or pentagonal dodecahedron these indices are 2, 1 + sf 5, 0. Kepler described as follows, briefly but adequately, the common characteristics of the dodecahedron and icosahedron : “Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quinquangulis figuratur expresse, hoc triangulis quidem sed in quinquanguli formam coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli structura perfici non potest sine proportione illa, quam hodierni geometrae divinam appellant” (De nive sexangula (1611), Opera, ed. Frisch, vi, p.723). Here Kepler was dealing, somewhat after the manner of Sir Thomas Browne, with the mysteries of the quincunx, and also of the hexagon; and was seeking for an explanation of the mysterious or even mystical beauty of the 5-petalled or 3-petalled flower,—pulehritudinis aut proprietatis figurae, quae animam harum plantarum characterisavit.

to a seriés 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 Miiller, and has ever since been used and quoted as Miiller’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

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.

Miiller’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 Miiller’s law, but as a second law applicable to a few particular cases. I on the other hand can find no case to which Miiller’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 Miiller’s law; for the four equatorial rays of Miiller (c) now correspond to the two polar and to two opposite equatorial facets of our polyhedron: the four “polar” rays of Miiller (a or e) correspond to two adjacent hexagons and two intermediate pentagons of the figure: and Miiller’s “tropical” rays (6 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.

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 radi. This figure has eighteen faces (F), twenty corners (C), and therefore thirty-six edges (#), 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, bedc, 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 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 te 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*; 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 * Cf. Tutton, Crystallography, p. 932, 1911. 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 geometrie, 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 proto-— plasmic 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

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. We have made use in the last chapter of the mathematical principle of Geodetics (or Geodesics) in order to explain the conformation of a certain class of sponge-spicules; but the principle is of much wider application in morphology, and would seem to deserve attention which it has not yet received.

Defining, meanwhile, our geodetic line (as we have already done) as the shortest distance between two points on the surface of a solid of revolution, we find that the geodetics of the cylinder give us one of the simplest of cases. Here it is plain that the geodetics are of three kinds: (1) a series of annuli around the cylinder, that is to say, a system of circles, in planes parallel to one another and at right angles to the axis of the cylinder (Fig. 236, a); (2) a series of straight lines parallel to the axis; and (3) a series of spiral curves winding round the wall of the cylinder

(b, c). These three systems are : B S ll of frequent occurrence, and Fig. 236. Annular and spiral thick- : 4 : ; enings in the walls of plant-cells. are all illustrated in the local thickenings of the wall of the cylindrical cells or vessels of plants. The spiral, or rather helicoid, geodetic is particularly common in cylindrical structures, and is beautifully shewn for instance in the spiral coil which stiffens the tracheal tubes of an insect, or the so-called “tracheides” of a woody stem. A similar pheno-

menon is often witnessed in the splitting of a glass tube. If a crack appear in a thin tube, such as a test-tube, it has a tendency to be prolonged in its own direction, and the more perfectly homogeneous and isotropic be the glass the more evenly will the split tend to follow the straight course in which it began. As a result, the crack in our test-tube is often seen to continue till the tube is split into a continuous spiral ribbon. In a right cone, the spiral geodetic falls into closer and closer coils as the diameter of the cone narrows; and a very beautiful geodetic of this kind is exemplified in the sutural line of a spiral shell, such as Turritella, or in the striations which run parallel with the spiral suture. Similarly, in an ellipsoidal surface, we have a spiral geodetic, whose coils get closer together as we approach the ends of the long axis of the ellipse; in the splitting of the integument of. an Equisetum-spore, by which are formed the spiral “elaters” of the spore, we have a case of this kind, though the spiral is not sufficiently prolonged to shew all its features in detail.

We have seen in these various cases, that our original definition of a geodetic requires to be modified; for it is only subject to conditions that it is “the shortest distance between two points on the surface of the solid,” and one of the commonest of these restricting conditions is that our geodetic may be constrained to go twice, or many times, round the surface on its way. In short, we must redefine our geodetic, as a curve drawn upon a surface, such that, if we take any two adjacent points on the curve, the curve gives the shortest distance between them. Again, in the geodetic systems which we meet with in morphology, it sometimes happens that we have two opposite systems of geodetic spirals separate and distinct from one another, as in Fig. 236, c; and it is also common to find the two systems interfering with one another, and forming a criss-cross, or reticulated arrangement. This is a very common source of reticulated patterns.

Among the ciliated Infusoria, we have in the spiral lines along which their cilia are arranged a great variety of beautiful geodetic curves; though it is probable enough that in some complicated cases these are not simple geodetics, but projections of curves other than a straight line upon the surface of the solid. Lastly, a very instructive case is furnished by the arrangement of the muscular fibres on the surface of a hollow organ, such as the heart or the stomach. Here we may consider the phenomenon from the point of view of mechanical efficiency, as well as from that of purely descriptive or objective anatomy. In fact we have an a priord right to expect that the muscular fibres covering such hollow or tubular organs will coincide with geodetic lines, in the sense in which we are now using the term. For if we imagine a contractile fibre, or elastic band, to be fixed by its two ends upon a curved surface, it is obvious that its first effort of contraction will tend to expend itself in accommodating the band to the form of the surface, in “stretching it tight,” or in other words in causing it to assume a direction which is the shortest possible line wpon the surface between the two extremes: and it is only then that further contraction will have the effect of constricting the tube and so exercising pressure on its contents. Thus the muscular fibres, as they wind over the curved surface of an organ, arrange themselves automatically in geodesic curves: in precisely the same manner as we also automatically construct complex systems of geodesics whenever we wind a ball of wool or a spindle of tow, or when the skilful surgeon bandages a limb. In these latter cases we see the production of those “figures-of-eight,” to which, in the case for instance of the heart-muscles, Pettigrew and other anatomists have ascribed peculiar importance. In the case of both heart and stomach we must look upon these organs as developed from a simple cylindrical tube, after the fashion of the glass-blower, as is further discussed on p. 737 of this book, the modification of the simple cylinder consisting of various degrees of dilatation and of twisting.