On Growth and Form

protoplasm of the cell on the other: and this involves a difference of tensions, so that the outer walls and their adjacent partitions are no longer likely to meet at equal angles of 120°. Moreover, a chemical change, due for instance to oxidation or possibly also to adsorption, is very likely to affect the external wall, and may tend to its consolidation; and this process, as we have seen, is tantamount to a large increase, and at the same time an equalisation, of tension in that outer wall, and will lead the adjacent partitions to impinge upon it at angles more and more nearly approximating to 90°: the bubble-like, or spherical, surfaces of the individual cells beg more and more flattened in consequence. Lastly, the chemical changes which affect the outer walls of the superficial cells may extend, in greater or less degree, to their inner walls also: with the result that these

cells will tend to become more or less rectangular throughout, and will cease to dovetail into the interstices of the next subjacent layer. These then are the general characters which we recognise in an epidermis; and we perceive that the fundamental character of an epidermis simply is that it les on the outside, and that its “main physical characteristics follow, as a matter of course, from the position which it occupies and from the various consequences which that situation entails. We have however by no means exhausted the subject in this short account; for the botanist is accustomed to draw a sharp distinction between a true epidermis and what is called epidermal tissue. The latter, which is found in such a sea-weed as Laminaria and in very many other cryptogamic plants, consists, as in the hypothetical case we have described, of a more or less simple and direct modification of the general or fundamental tissue. But a “true epidermis,” such as we have it in the higher plants, is something with a long morphological history, something which has been laid down or differentiated in an early stage of the plant’s growth, and which afterwards retains its separate and independent character. We shall see presently that a physical reason is again at hand to account, under certain circumstances, for the early partitioning off, from a mass of embryonic tissue, of an outer layer of cells which from their first appearance are marked off from the rest by their rectangular and flattened form.

We have hitherto considered our cells, or bubbles, as lying in a plane of symmetry, and further, we have only considered the appearance which they present as projected on that plane: in simpler words, we have been considering their appearance in surface or in sectional view. But we have further to consider them as solids, whether they be still grouped in relation to a single plane (like the four cells in Fig. 116) or heaped upon one another, as for instance in a tetrahedral form like four cannon-balls; and in either case we have to pass from the problems of plane to those of solid geometry. In short, the further development of our theme must lead us along two paths of enquiry, which continually intercross, namely (1) the study of more complex cases of partition and of contact in a plane, and (2) the whole question of the surfaces

and angles presented by solid figures in symmetrical juxtaposition. Let us take a simple case of the latter kind, and again afterwards, so far as possible, let us try to keep the two themes separate. Where we have three spheres in contact, as in Fig. 114 or in either half of Fig. 116, B, let us consider the point of contact (O, Fig. 114) not as a point in the plane section of the diagram, but as a point where three furrows meet on the surface of the system. At this point, three cells meet; but it is also obvious that there meet here six surfaces, namely the outer, spherical walls of the three bubbles, and the three partition-walls which divide them, two and two. Also, four lines or edges meet here; viz. the three external arcs which form the outer boundaries of the partition-walls (and which correspond to what we commonly call the “furrows” in the segmenting egg); and as a fourth edge, the “arris” or junction of the three partitions (perpendicular to the plane of the paper), where they all three meet together, as we have seen, at equal angles of 120°. Lastly, there meet at the point fowr solid angles, each bounded by three surfaces: to wit, within each bubble a solid angle bounded by two partition-walls and by the surface wall; and (fourthly) an external solid angle bounded by the outer surfaces of all three bubbles. Now in the case of the soap-bubbles (whose surfaces are all in contact with air, both outside and in), the six films meeting at the point, whether surface films or partition films, are all similar, with similar tensions. In other words the tensions, or forces, acting at the point are all similar and symmetrically arranged, and it at once follows from this that the angles, solid as well as plane, are all equal.

It is also obvious that, as regards the point of contact, the system will still be symmetrical, and its symmetry will be quite unchanged, if we add a fourth bubble in contact with the other three: that is to say, 1f where we had merely the outer air before, we now replace it by the air in the interior of another bubble. The only difference will be that the pressure exercised by the walls of this fourth bubble will alter the curvature of the surfaces of the others, so far as it encloses them; and, if all four bubbles be identical in size, these surfaces which formerly we called external and which have now come to be internal partitions, will, like the others, be flattened by equal and opposite pressure, into planes. We are now dealing, in short,

with six planes, meeting symmetrically in a point, and constituting there four equal solid angles. If we make a wire cage, in the form of a regular tetrahedron, and dip it into soap-solution, then when we withdraw it we see that to each one of the six edges of the tetrahedron, i.e. to each one of the six wires which constitute the little cage, a film has attached itself; and these six films meet internally at a point, and constitute in every respect the symmetrical figure which we have _ just been describing. In short, the system of films we have hereby automatically produced is precisely the system of partitionwalls which exist in our tetrahedral aggregation of four spherical

bubbles :—precisely the same, that is to say, in the neighbourhood of the meeting-point, and only differing in that we have made the wires of our tetrahedron straight, instead of imitating the circular arcs which actually form the intersections of our bubbles. ‘This detail we can easily introduce in our wire model if we please. Let us look for a moment at the geometry of our figure. Leto (Fig. 120) be the centre of the tetrahedron, i.e. the centre of symmetry where our films meet; and let oa, ob, oc, od, be lines drawn to the four corners of the tetrahedron. Produce ao to meet the base in p; then apd is a right-angled triangle. It is not difficult to prove that in such a figure, 0 (the centre of gravity of the system)

lies just three-quarters of the way between an apex, a, and a point, p, which is the centre of gravity of the opposite base. Therefore That is to say, the angle aod is just, as nearly as possible, 109° 28’ 16’. This angle, then, of 109° 28’ 16’, or very nearly 109 degrees and a half, is the angle at which, in this and every other solid system of liquid films, the edges of the partition-walls meet one another at a point. It is the fundamental angle in the solid geometry of our systems, just as 120° was the fundamental angle of symmetry so long as we considered only the plane projection, or plane section, of three films meeting in an edge.

Out of these two angles, we may construct a great variety of figures, plane and solid, which become all the more varied and complex when, by considering the case of unequal as well as equal cells, we admit curved (e.g. spherical) as well as plane boundary surfaces. Let us consider some examples and illustrations of these, beginning with those which we need only consider in reference to a plane. Let us imagine a system of equal cylinders, or equal spheres, in contact with one another in a plane, and represented in section by the equal and contiguous circles of Fig. 121. I borrow my figure, by the way, from an old Italian naturalist, Bonanni (a. contemporary of Borelh, of Ray and Willoughby and of Martin Lister), who dealt with this matter in a book chiefly devoted to molluscan shells*.

It is obvious, as a simple geometrical fact, that each of these equal circles is in contact with six surrounding circles. Imagine now that the whole system comes under some uniform stress. It may be of uniform surface tension at the boundaries of all the cells; it may be of pressure caused by uniform growth or expansion within the cells; or it may be due to some uniformly applied constricting pressure from without. In all of these cases the points of contact between the circles in the diagram will be extended into

* Ricreatione dell’ occhio e della mente, nell’ Osservatione delle Chiocciole, Roma, lines of contact, representing surfaces of contact in the actual spheres or cylinders; and the equal circles of our diagram will be converted into regular and equal hexagons. The angles of these hexagons, at each of which three hexagons meet, are of course angles of 120°. So far as the form is concerned, so long as we are concerned only with a morphological result and not with a physiological process, the result is precisely the same whatever be the force which brings the bodies together in symmetrical apposition ; it is by no means necessary for us, in the first instance, even to enquire whether it be surface tension or mechanical

pressure or some other physical force which is the cause, or the main cause, of the phenomenon. The production by mutual interaction of polyhedral cells, which, under conditions of perfect symmetry, become regular hexagons, is very beautifully illustrated by Prof. Bénard’s “tourbillons cellulaires”’ (cf. p. 259), and also in some of Leduc’s diffusion experiments. A weak (5 per cent.) solution of gelatine is allowed to set on a plate of glass, and little drops of a 5 or 10 per cent. solution of ferrocyanide of potassium are then placed at regular intervals upon the gelatine. Immediately each little drop becomes the centre, or pole, of a system of diffusion currents,

and the several systems conflict with and repel one another, so that presently each little area becomes the seat of a double current system, from its centre outwards and back again; until at length the concentration of the field becomes equalised and the currents Fig. 122. An “artificial tissue,” formed by coloured drops of sodium chloride solution diffusing in a less dense solution of the same salt. (After Leduc.) Fig. 123. An artificial cellular tissue, formed by the diffusion in gelatine of drops of a solution of potassium ferrocyanide. (After Leduc.)

cease. After equilibrium is attained, and when the gelatinous mass is permitted to dry, we have an artificial tissue of more or less regularly hexagonal “cells,” which simulate in the closest way an organic parenchyma. And by varying the experiment, in ways which Leduc describes, we may simulate various forms of tissue, and produce cells with thick walls or with thin, cells in close contact or with wide intercellular spaces, cells with plane or with curved partitions, and so forth.

The hexagonal pattern is illustrated among organisms in countless cases, but those in which the pattern is perfectly regular, by reason of perfect uniformity of force and perfect equality of the individual cells, are not so numerous. The hexagonal epitheliumcells of the pigment layer of the eye, external to the retina, are a good example. Here we have a single layer of uniform cells, reposing on the one hand upon a basement membrane, supported

behind by the solid wall of the sclerotic, and exposed on the other hand to the uniform fluid pressure of the vitreous humour. The conditions all point, and lead, to a perfectly symmetrical result : that is to say, the cells, uniform in size, are flattened out to a uniform thickness by the fluid pressure acting radially; and their reaction on each other converts the flattened discs into regular hexagons. In an ordinary columnar epithelium, such as that of the intestine, we see again that the columnar cells have been compressed into hexagonal prisms; but here as a rule the cells are less uniform in size, small cells are apt to be intercalated among the larger, and the perfect symmetry is accordingly lost. The same is true of ordinary vegetable parenchyma; the originally spherical cells are approximately equal in size, but only approximately ; and there are accordingly all degrees in the regularity and symmetry of the resulting tissue. But obviously, wherever we

have, in addition to the forces which tend to produce the regular hexagonal symmetry, some other asymmetrical component arising from growth or traction, then our regular hexagons will be distorted in various simple ways. This condition is illustrated in the accompanying diagram of the epidermis of Girardia; it also accounts for the more or less pointed or fusiform cells, each still in contact (as a rule) with six others, which form the epithelial lining of the blood-vessels: and other similar, or analogous, instances are very common.

In a soap-froth imprisoned between two glass plates, we have a symmetrical system of cells, which appear in optical section (as in Fig. 125, B) as regular hexagons; but if we press the plates a little closer together, the hexagons become deformed or flattened (Fig. 125, A). In this case, however, if we cease to apply further pressure, the tension of the films throughout the system soon adjusts itself again, and in a short time the system has regained the former symmetry of Fig. 125, B.

In the growth of an ordinary dicotyledonous leaf, we once more see reflected in the form of its epidermal cells the tractions, irregular but on the whole longitudinal, Fig. 126. From leaf of Which growth has superposed on the ten- Elodea canadensis. (After sions of the partition-walls (Fig. 126). In el) the narrow elongated leaf of a Monocotyledon, such as a hyacinth, the elongated, apparently quadrangular cells of the epidermis appear as a necessary consequence of the simpler laws of growth which gave its simple form to the leaf as a whole. In this last case, however, as in all the others, the rule still holds that only three partitions (in surface view) meet in a point; and at their point of meeting the walls are for a short distance manifestly curved, so as to permit the junction to take place at or nearly at the normal angle of 120°.

Briefly speaking, wherever we have a system of cylinders or spheres, associated together with sufficient mutual interaction to bring them into complete surface contact, there, in section or in surface view, we tend to get a pattern of hexagons. While the formation of an hexagonal pattern on the basis of ready-formed and symmetrically arranged material units is a very common, and indeed the general way, it does not follow that there are not others by which such a pattern can be obtained. For instance, if we take a little triangular dish of mercury and set it vibrating (either by help of a tuning-fork, or by simply tapping on the sides) we shall have a series of little waves or ripples starting inwards from each of the three faces; and the intercrossing, or interference of these three sets of waves produces crests and hollows, and intermediate points of no disturbance, whose loci are seen as a beautiful pattern of minute hexagons. It is possible that the very minute and astonishingly regular pattern of hexagons which we see, for instance, on the surface of many diatoms, may be a phenomenon of this order*. The same may be the case also in Arcella, where an apparently hexagonal pattern is found not to consist of simple hexagons, but of “straight lines in three sets of parallels, the lines of each set making an angle of sixty degrees with those of the other two sets{.” We must also bear in mind, in the case of the minuter forms, the large possibilities of optical illusion. For instance, in one of Abbe’s “‘diffraction-plates,” a pattern of dots, set at equal interspaces, is reproduced on a very minute scale by photography; but under certain conditions of microscopic illumination and focussing, these isolated dots appear as a pattern of hexagons.

A symmetrical arrangement of hexagons, such as we have just been studying, suggests various simple geometrical corollaries, of which the following may perhaps be a useful one. We may sometimes desire to estimate the number of hexagonal areas or facets in some structure where these are numerous, such for instance as the * Cf. some of J. H. Vincent’s photographs of ripples, in Phil. Mag. 1897-1899; or those of F. R. Watson, in Phys. Review, 1897, 1901, 1916. The appearance will depend on the rate of the wave, and in turn on the surface-tension; with a low tension one would probably see only a moving “jabble.’’ FitzGerald thought diatom-patterns might be due to electromagnetic vibrations (Works, p. 503, 1902).

cornea of an insect’s eye, or in the minute pattern of hexagons on many diatoms. An approximate enumeration is easily made as follows. For the area of a hexagon (if we call 6 the short diameter, that namely which bisects two of the opposite sides) is 62 x 3/2, the area of a circle being d?. 7/4. Then, if the diameter (d) of a circular area include n hexagons, the area of that circle equals (n. 5)? x 7/4. And, dividing this number by the area of a single hexagon, we obtain for the number of areas in the circle, ' each equal to a hexagonal facet, the expression n2 x 2/4 x 2/s/3 = 0-907n2, or 9/10. ?, nearly.

This calculation deals, not only with the complete facets, but with the areas of the broken hexagons at the periphery of the circle. If we neglect these latter, and consider our whole field as consisting of successive rings of hexagons about a central one, we may obtain a still simpler rule*. For obviously, around our central hexagon there stands a zone of six, and around these a zone of twelve, and around these a zone of eighteen, and soon. And the total number, excluding the central hexagon, is accordingly:

and so forth. If NV be the number of zones, and if we add one to the above numbers for the odd central hexagon, the rule evidently is, that the total number, H, = 3N(N+1)+ 1. Thus, if in a preparation of a fly’s cornea, I can count twenty-five facets in a line from a central one, the total number in the entire circular field is (3 x 25 x 26) + 1= 195I1f. The same principles which account for the development of hexagonal symmetry hold true, as a matter of course, not only of individual cells (in the biological sense), but of any closepacked bodies of uniform size and originally circular outline; and the hexagonal pattern is therefore of very common occurrence, under widely different circumstances. The curious reader may consult Sir Thomas Browne’s quaint and beautiful account, in the Garden of Cyrus, of hexagonal (and also of quincuncial) symmetry in plants and animals, which “doth neatly declare how nature Geometrizeth, and observeth order in all things.”

* This does not merely neglect the broken ones but all whose centres lie between this circle and a hexagon inscribed in it. *We have many varied examples of this principle among corals, wherever the polypes are in close juxtaposition, with neither empty space nor accumulations of matrix between their adjacent walls. Favosites gothlandica, for instance, furnishes us with an excellent example. In the great genus Lithostrotion we have some species that are “massive” and others that are “fasciculate” ; in other words in some the long cylindrical corallites are in close contact with one another, and in others they are separate and loosely bundled (Fig. 127). Accordingly in the former the corallites are

Fig. 127. Lithostrotion Martini. Fig. 128. Cyathophyllum hexagonum. (After Nicholson.) (From Nicholson, after Zittel.) squeezed into hexagonal prisms, while in the latter they retain their cylindrical form. Where the polypes are comparatively few, and so have room to spread, the mutual pressure ceases to work or only tends to push them asunder, letting them remain circular in outline (e.g. Thecosmila). Where they vary gradually in size, as for instance in Cyathophyllum hexagonum, they are more or less hexagonal but are not regular hexagons; and where there is greater and more irregular variation in size, the cells will be on the average hexagonal, but some will have fewer and some more sides than six, as in the annexed figure of Arachnophyllum (Fig. 129).

Where larger and smaller cells, corresponding to two different kinds of zooids, are mixed together, we may get various results. If the larger cells are numerous enough to be more or less in contact with one another (e.g. various Monticuliporae) they will be irregular hexagons, while the smaller cells between them will be crushed into all manner of irregular angular forms. If on the other hand the large cells are comparatively few and are large and strong-walled compared with their smaller neighbours, then the latter alone will be squeezed into hexagons, while the larger ones will tend to retain their circular outline undisturbed (e.g. Heliopora, Heliolites, etc.).

“thecae” of the individual polypes remain undeveloped but the radiating septa are formed and calcified, then we obtain new and beautiful mathematical configurations (Fig. 131). For the radiating septa are no longer confined to the circular or hexagonal bounds of a polypite, but tend to meet and become confluent with their neighbours on every side; and, tending to assume positions of equilibrium, or of minimal area, under the restraints to which they are subject, they fall into congruent curves; and these correspond, in a striking manner, to the lines of force running, in a common field of force, between a number of secondary centres. Similar patterns may be produced in various ways, by the play of osmotic or magnetic forces; and a particular and very curious case is to be found in those complicated forms of nuclear division

known as triasters, polyasters, etc., whose relation to a field of force Hartog has explained*. It is obvious that, in our corals, these curving septa are all orthogonal to the non-existent hexagonal boundaries. As the phenomenon is wholly due to the imperfect development or non-existence of a thecal wall, it is not surprising that we find identical configurations among various corals, or families of corals, not otherwise related to one another; we find the same or very similar patterns displayed, for instance, in Synhelia (Oculinidae), in Phillipsastraea (Rugosa), in Thamnastraea (Fungida), and in many more.

The most famous of all hexagonal conformations and perhaps the most beautiful is that of the bee’s cell. Here we have, as in Fig. 131. Surface-views of Corals with undeveloped thecae and confluent septa. A, Thamnastraea; B, Comoseris. (From Nicholson, after Zittel.) our last examples, a series of equal cylinders, compressed by symmetrical forces into regular hexagonal prisms. But in this case we have two rows of such cylinders, set opposite to one another, end to end; and we have accordingly to consider also the conformation of their ends. We may suppose our original cylindrical cells to have spherical ends, which is their normal and symmetrical mode of termination; and, for closest packing, it is obvious that the end of any one cylinder will touch, and fit in between, the ends of three cylinders in the opposite row. It is just as when we pile round-shot in a heap; each.sphere that we