THE GEOMETRY OF THE EPEIRA’S WEB

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APPENDIX: THE GEOMETRY OF THE EPEIRA’S WEB

I find myself confronted with a subject which is not only highly interesting, but somewhat difficult: not that the subject is obscure; but it presupposes in the reader a certain knowledge of geometry: a strong meat too often neglected. I am not addressing geometricians, who are generally indifferent to questions of instinct, nor entomological collectors, who, as such, take no interest in mathematical theorems; I write for any one with sufficient intelligence to enjoy the lessons which the insect teaches.

What am I to do? To suppress this chapter were to leave out the most remarkable instance of Spider industry; to treat it as it should be treated, that is to say, with the whole armoury of scientific formulae, would be out of place in these modest pages. Let us take a middle course, avoiding both abstruse truths and complete ignorance.

Let us direct our attention to the nets of the Epeirae, preferably to those of the Silky Epeira and the Banded Epeira, so plentiful in the autumn, in my part of the country, and so remarkable for their bulk. We shall first observe that the radii are equally spaced; the angles formed by each consecutive pair are of perceptibly equal value; and this in spite of their number, which in the case of the Silky Epeira exceeds two score. We know by what strange means the Spider attains her ends and divides the area wherein the web is to be warped into a large number of equal sectors, a number which is almost invariable in the work of each species. An operation without method, governed, one might imagine, by an irresponsible whim, results in a beautiful rose-window worthy of our compasses.

We shall also notice that, in each sector, the various chords, the elements of the spiral windings, are parallel to one another and gradually draw closer together as they near the centre. With the two radiating lines that frame them they form obtuse angles on one side and acute angles on the other; and these angles remain constant in the same sector, because the chords are parallel.

There is more than this: these same angles, the obtuse as well as the acute, do not alter in value, from one sector to another, at any rate so far as the conscientious eye can judge. Taken as a whole, therefore, the rope-latticed edifice consists of a series of cross-bars intersecting the several radiating lines obliquely at angles of equal value.

By this characteristic we recognize the ‘logarithmic spiral.’ Geometricians give this name to the curve which intersects obliquely, at angles of unvarying value, all the straight lines or ‘radii vectores’ radiating from a centre called the ‘Pole.’ The Epeira’s construction, therefore, is a series of chords joining the intersections of a logarithmic spiral with a series of radii. It would become merged in this spiral if the number of radii were infinite, for this would reduce the length of the rectilinear elements indefinitely and change this polygonal line into a curve.

To suggest an explanation why this spiral has so greatly exercised the meditations of science, let us confine ourselves for the present to a few statements of which the reader will find the proof in any treatise on higher geometry.

The logarithmic spiral describes an endless number of circuits around its pole, to which it constantly draws nearer without ever being able to reach it. This central point is indefinitely inaccessible at each approaching turn. It is obvious that this property is beyond our sensory scope. Even with the help of the best philosophical instruments, our sight could not follow its interminable windings and would soon abandon the attempt to divide the invisible. It is a volute to which the brain conceives no limits. The trained mind, alone, more discerning than our retina, sees clearly that which defies the perceptive faculties of the eye.

The Epeira complies to the best of her ability with this law of the endless volute. The spiral revolutions come closer together as they approach the pole. At a given distance, they stop abruptly; but, at this point, the auxiliary spiral, which is not destroyed in the central region, takes up the thread; and we see it, not without some surprise, draw nearer to the pole in ever-narrowing and scarcely perceptible circles. There is not, of course, absolute mathematical accuracy, but a very close approximation to that accuracy. The Epeira winds nearer and nearer round her pole, so far as her equipment, which, like our own, is defective, will allow her. One would believe her to be thoroughly versed in the laws of the spiral.

I will continue to set forth, without explanations, some of the properties of this curious curve. Picture a flexible thread wound round a logarithmic spiral. If we then unwind it, keeping it taut the while, its free extremity will describe a spiral similar at all points to the original. The curve will merely have changed places.

Jacques Bernouilli, {42} to whom geometry owes this magnificent theorem, had engraved on his tomb, as one of his proudest titles to fame, the generating spiral and its double, begotten of the unwinding of the thread. An inscription proclaimed, ‘Eadem mutata resurgo: I rise again like unto myself.’ Geometry would find it difficult to better this splendid flight of fancy towards the great problem of the hereafter.

There is another geometrical epitaph no less famous. Cicero, when quaestor in Sicily, searching for the tomb of Archimedes amid the thorns and brambles that cover us with oblivion, recognized it, among the ruins, by the geometrical figure engraved upon the stone: the cylinder circumscribing the sphere. Archimedes, in fact, was the first to know the approximate relation of circumference to diameter; from it he deduced the perimeter and surface of the circle, as well as the surface and volume of the sphere. He showed that the surface and volume of the last-named equal two-thirds of the surface and volume of the circumscribing cylinder. Disdaining all pompous inscription, the learned Syracusan honoured himself with his theorem as his sole epitaph. The geometrical figure proclaimed the individual’s name as plainly as would any alphabetical characters.

To have done with this part of our subject, here is another property of the logarithmic spiral. Roll the curve along an indefinite straight line. Its pole will become displaced while still keeping on one straight line. The endless scroll leads to rectilinear progression; the perpetually varied begets uniformity.

Now is this logarithmic spiral, with its curious properties, merely a conception of the geometers, combining number and extent, at will, so as to imagine a tenebrous abyss wherein to practise their analytical methods afterwards? Is it a mere dream in the night of the intricate, an abstract riddle flung out for our understanding to browse upon?

No, it is a reality in the service of life, a method of construction frequently employed in animal architecture. The Mollusc, in particular, never rolls the winding ramp of the shell without reference to the scientific curve. The first-born of the species knew it and put it into practice; it was as perfect in the dawn of creation as it can be to-day.

Let us study, in this connection, the Ammonites, those venerable relics of what was once the highest expression of living things, at the time when the solid land was taking shape from the oceanic ooze. Cut and polished length-wise, the fossil shows a magnificent logarithmic spiral, the general pattern of the dwelling which was a pearl palace, with numerous chambers traversed by a siphuncular corridor.

To this day, the last representative of the Cephalopoda with partitioned shells, the Nautilus of the Southern Seas, remains faithful to the ancient design; it has not improved upon its distant predecessors. It has altered the position of the siphuncle, has placed it in the centre instead of leaving it on the back, but it still whirls its spiral logarithmically as did the Ammonites in the earliest ages of the world’s existence.

And let us not run away with the idea that these princes of the Mollusc tribe have a monopoly of the scientific curve. In the stagnant waters of our grassy ditches, the flat shells, the humble Planorbes, sometimes no bigger than a duckweed, vie with the Ammonite and the Nautilus in matters of higher geometry. At least one of them, Planorbis vortex, for example, is a marvel of logarithmic whorls.

In the long-shaped shells, the structure becomes more complex, though remaining subject to the same fundamental laws. I have before my eyes some species of the genus Terebra, from New Caledonia. They are extremely tapering cones, attaining almost nine inches in length. Their surface is smooth and quite plain, without any of the usual ornaments, such as furrows, knots or strings of pearls. The spiral edifice is superb, graced with its own simplicity alone. I count a score of whorls which gradually decrease until they vanish in the delicate point. They are edged with a fine groove.

I take a pencil and draw a rough generating line to this cone; and, relying merely on the evidence of my eyes, which are more or less practised in geometric measurements, I find that the spiral groove intersects this generating line at an angle of unvarying value.

The consequence of this result is easily deduced. If projected on a plane perpendicular to the axis of the shell, the generating lines of the cone would become radii; and the groove which winds upwards from the base to the apex would be converted into a plane curve which, meeting those radii at an unvarying angle, would be neither more nor less than a logarithmic spiral. Conversely, the groove of the shell may be considered as the projection of this spiral on a conic surface.

Better still. Let us imagine a plane perpendicular to the aids of the shell and passing through its summit. Let us imagine, moreover, a thread wound along the spiral groove. Let us unroll the thread, holding it taut as we do so. Its extremity will not leave the plane and will describe a logarithmic spiral within it. It is, in a more complicated degree, a variant of Bernouilli’s ‘Eadem mutata resurgo:’ the logarithmic conic curve becomes a logarithmic plane curve.

A similar geometry is found in the other shells with elongated cones, Turritellae, Spindle-shells, Cerithia, as well as in the shells with flattened cones, Trochidae, Turbines. The spherical shells, those whirled into a volute, are no exception to this rule. All, down to the common Snail-shell, are constructed according to logarithmic laws. The famous spiral of the geometers is the general plan followed by the Mollusc rolling its stone sheath.

Where do these glairy creatures pick up this science? We are told that the Mollusc derives from the Worm. One day, the Worm, rendered frisky by the sun, emancipated itself, brandished its tail and twisted it into a corkscrew for sheer glee. There and then the plan of the future spiral shell was discovered.

This is what is taught quite seriously, in these days, as the very last word in scientific progress. It remains to be seen up to what point the explanation is acceptable. The Spider, for her part, will have none of it. Unrelated to the appendix-lacking, corkscrew-twirling Worm, she is nevertheless familiar with the logarithmic spiral. From the celebrated curve she obtains merely a sort of framework; but, elementary though this framework be, it clearly marks the ideal edifice. The Epeira works on the same principles as the Mollusc of the convoluted shell.

The Mollusc has years wherein to construct its spiral and it uses the utmost finish in the whirling process. The Epeira, to spread her net, has but an hour’s sitting at the most, wherefore the speed at which she works compels her to rest content with a simpler production. She shortens the task by confining herself to a skeleton of the curve which the other describes to perfection.

The Epeira, therefore, is versed in the geometric secrets of the Ammonite and the Nautilus pompilus; she uses, in a simpler form, the logarithmic line dear to the Snail. What guides her? There is no appeal here to a wriggle of some kind, as in the case of the Worm that ambitiously aspires to become a Mollusc. The animal must needs carry within itself a virtual diagram of its spiral. Accident, however fruitful in surprises we may presume it to be, can never have taught it the higher geometry wherein our own intelligence at once goes astray, without a strict preliminary training.

Are we to recognize a mere effect of organic structure in the Epeira’s art? We readily think of the legs, which, endowed with a very varying power of extension, might serve as compasses. More or less bent, more or less outstretched, they would mechanically determine the angle whereat the spiral shall intersect the radius; they would maintain the parallel of the chords in each sector.

Certain objections arise to affirm that, in this instance, the tool is not the sole regulator of the work. Were the arrangement of the thread determined by the length of the legs, we should find the spiral volutes separated more widely from one another in proportion to the greater length of implement in the spinstress. We see this in the Banded Epeira and the Silky Epeira. The first has longer limbs and spaces her cross-threads more liberally than does the second, whose legs are shorter.

But we must not rely too much on this rule, say others. The Angular Epeira, the Paletinted Epeira and the Cross Spider, all three more or less short-limbed, rival the Banded Epeira in the spacing of their lime-snares. The last two even dispose them with greater intervening distances.

We recognize in another respect that the organization of the animal does not imply an immutable type of work. Before beginning the sticky spiral, the Epeirae first spin an auxiliary intended to strengthen the stays. This spiral, formed of plain, non-glutinous thread, starts from the centre and winds in rapidly-widening circles to the circumference. It is merely a temporary construction, whereof naught but the central part survives when the Spider has set its limy meshes. The second spiral, the essential part of the snare, proceeds, on the contrary, in serried coils from the circumference to the centre and is composed entirely of viscous cross-threads.

Here we have, following one after the other merely by a sudden alteration of the machine, two volutes of an entirely different order as regards direction, the number of whorls and intersection. Both of them are logarithmic spirals. I see no mechanism of the legs, be they long or short, that can account for this alteration.

Can it then be a premeditated design on the part of the Epeira? Can there be calculation, measurement of angles, gauging of the parallel by means of the eye or otherwise? I am inclined to think that there is none of all this, or at least nothing but an innate propensity, whose effects the animal is no more able to control than the flower is able to control the arrangement of its verticils. The Epeira practises higher geometry without knowing or caring. The thing works of itself and takes its impetus from an instinct imposed upon creation from the start.

The stone thrown by the hand returns to earth describing a certain curve; the dead leaf torn and wafted away by a breath of wind makes its journey from the tree to the ground with a similar curve. On neither the one side nor the other is there any action by the moving body to regulate the fall; nevertheless, the descent takes place according to a scientific trajectory, the ‘parabola,’ of which the section of a cone by a plane furnished the prototype to the geometer’s speculations. A figure, which was at first but a tentative glimpse, becomes a reality by the fall of a pebble out of the vertical.

The same speculations take up the parabola once more, imagine it rolling on an indefinite straight line and ask what course does the focus of this curve follow. The answer comes: The focus of the parabola describes a ‘catenary,’ a line very simple in shape, but endowed with an algebraic symbol that has to resort to a kind of cabalistic number at variance with any sort of numeration, so much so that the unit refuses to express it, however much we subdivide the unit. It is called the number e. Its value is represented by the following series carried out ad infinitum:

e = 1 + 1/1 + 1/(1*2) + 1/(1*2*3) + 1/(1*2*3*4) + 1/(1*2*3*4*5) + etc

If the reader had the patience to work out the few initial terms of this series, which has no limit, because the series of natural numerals itself has none, he would find:

e=2.7182818...

With this weird number are we now stationed within the strictly defined realm of the imagination? Not at all: the catenary appears actually every time that weight and flexibility act in concert. The name is given to the curve formed by a chain suspended by two of its points which are not placed on a vertical line. It is the shape taken by a flexible cord when held at each end and relaxed; it is the line that governs the shape of a sail bellying in the wind; it is the curve of the nanny-goat’s milk-bag when she returns from filling her trailing udder. And all this answers to the number e.

What a quantity of abstruse science for a bit of string! Let us not be surprised. A pellet of shot swinging at the end of a thread, a drop of dew trickling down a straw, a splash of water rippling under the kisses of the air, a mere trifle, after all, requires a titanic scaffolding when we wish to examine it with the eye of calculation. We need the club of Hercules to crush a fly.

Our methods of mathematical investigation are certainly ingenious; we cannot too much admire the mighty brains that have invented them; but how slow and laborious they appear when compared with the smallest actualities! Will it never be given to us to probe reality in a simpler fashion? Will our intelligence be able one day to dispense with the heavy arsenal of formulae? Why not?

Here we have the abracadabric number e reappearing, inscribed on a Spider’s thread. Let us examine, on a misty morning, the meshwork that has been constructed during the night. Owing to their hygrometrical nature, the sticky threads are laden with tiny drops, and, bending under the burden, have become so many catenaries, so many chaplets of limpid gems, graceful chaplets arranged in exquisite order and following the curve of a swing. If the sun pierce the mist, the whole lights up with iridescent fires and becomes a resplendent cluster of diamonds. The number e is in its glory.

Geometry, that is to say, the science of harmony in space, presides over everything. We find it in the arrangement of the scales of a fir-cone, as in the arrangement of an Epeira’s limy web; we find it in the spiral of a Snail-shell, in the chaplet of a Spider’s thread, as in the orbit of a planet; it is everywhere, as perfect in the world of atoms as in the world of immensities.

And this universal geometry tells us of an Universal Geometrician, whose divine compass has measured all things. I prefer that, as an explanation of the logarithmic curve of the Ammonite and the Epeira, to the Worm screwing up the tip of its tail. It may not perhaps be in accordance with latter-day teaching, but it takes a loftier flight.

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