INSECT GEOMETRY
The industry of insects, especially that of the Bees and Wasps, abounds
in tiny marvels. Newly manufactured with the cotton supplied by various
fluff-covered plants, the nest of certain Anthidia forms an exquisitely
graceful pouch. It is accurately fashioned, white as snow, pleasing to
the eye and softer to the touch than Swan’s-down. The Humming-bird’s
nest, a bowl hardly half the size of an apricot, is by comparison a
piece of clumsy felt.
But this perfection is of brief duration. The artist is hampered by the
exigencies of the space at her disposal. Her workshop is a chance
shelter, a tunnel incapable of modification, which she has to use as
she finds it. In this narrow retreat, therefore, the cotton purses are
placed in a row, each compressing the others and distorting their form;
they are welded at either end to their neighbours, till the whole
becomes a lumpy pillar moulded to the volume of the container. For lack
of space, the weaver has been unable to continue her textile fabric in
accordance with the exquisite design dictated by her instinct. A length
of rope, of indifferent merit, takes the place of the superb
masterpiece of felt which the Anthidium would have created had she been
working at isolated cells.
The Chalicodoma of the Walls, when building on a pebble, first raises a
turret of faultless geometrical proportions. The dust scraped from the
hardest spots in the highways and kneaded with saliva provides the
mortar. To make a more solid job of things and also to economize
cement, which takes a long time to collect and prepare, tiny bits of
gravel are encrusted in the outer surface before the material sets. In
this way the initial building becomes a rustic rockwork fortress, which
is quite pretty to look at.
Using her trowel freely, the Mason-bee has builded after the prototype
of her art, the cylinder adorned with a mosaic pattern. But other
cells, at least a dozen, are to follow. Necessities now obtrude
themselves from which the first piece of work was exempt; that which
will soon be building is subordinated to that which is already built.
The solidity of the whole requires that the turrets leaning one against
the other shall form a solid mass; and economy of material demands that
the same partition-wall shall serve for two adjoining cells. These two
conditions are incompatible with the regulation architecture, for
grouped cylinders touch only along a line, affording no appreciable
area of common partition-wall; they leave between them unoccupied
intervals, which would prejudice the general stability. What does the
builder do to remedy these two defects?
She abandons the normal outline and modifies it according to the space
at her disposal. She alters the shape of the cylinder, not as regards
the interior, which is still kept rounded to suit the convenience of
the larva, the future inhabitant, but as regards the outer envelope,
which becomes irregular and polygonal, filling the interstices with its
angles.
The exquisite geometry promised by the turret first constructed is
perforce abandoned when the complete edifice has to consist of a mass
of cells in juxtaposition. Inexactness follows exactness even more
noticeably at the end of the task. Anxious to strengthen her work and
enable it to resist the attacks of the weather, the mason plasters it
with a thick layer of mortar. Mosaic encrustations, round mouths,
closed with a lid, and cylindrical bastions: all these disappear,
submerged by the defensive casing. To look at, there is nothing left
but a clod of dried mud.
The simplest of round bodies, the cylinder, stands likewise as the
model for the jam-pot wherein the Pelopæus stacks her Spiders. With mud
collected from the edge of a pool, the huntress begins by building a
turret ornamented with diagonal lozenges. Unhampered by its
surroundings, this structure, the first of the group, is of a
perfection that gives us a high opinion of the builder’s talent. It is
fashioned like a segment of a twisted column. But other cells follow
which, leaning one against the other, produce a mutual distortion. For
the same reasons, namely, economy of material and general solidity, the
beautiful ordonnance promised at the outset is wanting; crowding leads
to irregularity. A thick layer of cement ends by deforming the
structure altogether.
Let us next consider the Agenia, who rivals the Pelopæus as a huntress
and a worker in clay. She encloses the one Spider who forms her larva’s
ration in an earthenware shell hardly as large as a cherry-stone and
embellished on the outside with a tiny milled pattern. This little gem
of ceramics is an ellipsoid truncated at one end. When the structure
stands alone, its accuracy of form is perfect.
But the potter’s ware does not end with this. The place of refuge
discovered in some crevice in a sunny wall is a valuable site, where
the whole family will take up its abode. More preserve-jars are
therefore fashioned, sometimes arranged in a row, sometimes collected
in a group. Though constructed according to a fixed type, the
ellipsoid, the new structures depart, some more, some less, from the
ideal model. Welded together, end to end, they lose the smooth nipple
of the ellipse and replace it by the sudden truncation of the barrel.
When they are joined lengthwise, the belly of the barrel becomes
flattened; when they are massed together anyhow, they become almost
unrecognizable. Nevertheless, as the Agenia, unlike the Pelopæus, never
covers her collection of pots with a casing, her work retains its
distinctive features fairly well, thanks to the thoroughness with which
the artist has stamped her trade-mark upon it.
The pottery of the Eumenes is of a higher order: it favours a bulging
cupola, like that of the Turkish kiosk or the Moscow basilica. At the
summit of the dome is a short opening, like the mouth of an amphora,
through which the caterpillars intended for the larva’s consumption are
introduced. When the larder is full and the egg slung from the ceiling
by a thread, the bell-mouthed neck of the cell is closed with a clay
stopper.
As a rule, in these parts, E. Amadei builds on a big pebble. She adorns
her cupola with angular bits of gravel, half buried in the plaster; on
the stopper closing the mouth she places a little flat stone, or even a
Snail-shell, selected from among the smallest. The earthenware
casemate, well-baked by the sun, is supremely graceful.
Well, this elegant structure is doomed to disappear. Around her cupola
the Eumenes builds others, using as walls what she has already built.
Henceforth the exact circular form is no longer practicable. In order
to occupy the reentrant angles, the new cells themselves become angular
and assume an undecided, polyhedral form. Only the edges of the mass
and the top retain traces of the regulation plan. The nest as a whole
shows a nippled surface encrusted with broken flint. Each nipple
corresponds with a cell, which may always be known by its amphora-like
mouth, a part which is not misshapen, because it has been fashioned
without impediment. In the absence of this certificate of origin, we
should hesitate before recognizing the work of an expert dome-builder
in the shapeless blob.
E. unguiculata does worse. After building, on some big stone, a group
of cells which, in shape, ornamental encrustation and bell-mouthed
neck, rival those of E. Amadei, she buries the whole under a layer of
mortar. She imitates the Chalicodoma and the Pelopæus, who, for reasons
of domestic safety, follow up artistic daintiness with the uncouthness
of the fortress. Inspired by a system of æsthetics which nothing is
able to evade, both insects begin by creating beauty; dominated by the
fear of danger, they end by creating ugliness.
Other Eumenes, on the contrary, of smaller size, build cells which are
always isolated and which often have the twig of a shrub for a support.
The structure is a cupola, similar to those already mentioned, and,
like them, provided with a graceful neck, but without the gravel
mosaic. The tiny fabric, no bigger than a cherry, does not admit of
this rustic ornamentation. The potter replaces it by a few specks of
clay distributed here and there.
The Eumenes who build a succession of cells in groups are compelled to
deform the chamber under construction according to the space left by
those preceding it; for the beautiful curve of their original design
they substitute, by force of circumstances, the unpleasing broken line.
The others, those who build each cell in isolation, are far from
perpetrating such inaccuracies. From first to last, as many as the
establishment of the larvæ requires, now on this twig, now on that, the
cells are built of an identical shape, just as though they had issued
from the same mould. Now that nothing hinders the exact application of
the rules, order returns and produces a series of structures which are
no less perfect at the end than at the beginning.
If the insect were to build a general shelter, in which each larva had
its individual box, what would this building, this common home of the
family, be? On condition, of course, that no obstacle intervene, the
work will always be correct in its geometry, which will vary according
to the builder’s speciality. I could draw you a child’s balloon than
which none prettier was ever inflated in toyland, or, for that matter,
in fairyland; and it would be exactly like the nest of a Median Wasp
(Vespa media, De Geer). The person who gave me this marvel found it
hanging from the lower edge of a shutter which was left open for the
greater part of the year.
Possessing liberty of action in all directions, except at the point of
contact with the shutter, the Wasp followed the rules of her art
without impediment. With a paper of her own manufacture, tough and
flexible as the silk papers of China and Japan, she contrived to expand
her work into a segment of an ellipsoid, with a cone added to it by
means of a gentle curve. A like association of forms artistically
combined is found in the Sacred Beetle’s pears. [62] The slender Wasp
and the heavy Dung-beetle, employing dissimilar tools and materials,
work after the same pattern.
Ill-defined spiral bands tell us how the Wasp went to work. With her
pellet of paper-pulp in her mandibles, she moved downwards in a
slanting direction, following the margin of the part already
constructed and leaving as she went a ribbon of her material, still
quite soft and impregnated with saliva. The work was discontinued and
resumed hundreds and hundreds of times, for the supply was soon
exhausted. The Wasp had to go to some woody stem hard by, a stem retted
by the moist air and bleached by the sun, and scrape it with her teeth;
she had to tear out its fibres, to divide them, unravel them and work
them up into a plastic felt. When the pellet was removed, the Wasp
hastened back to resume her interrupted ribbon.
There was even the collaboration of several builders. The foundress of
the city, the mother, alone at the outset and absorbed by family cares,
was able only to make a rough beginning of the roof; but offspring
arrived, neuters, [63] eager assistants henceforth charged with the
continuing and enlarging of the dwelling, in order to provide the one
mother with a lodging to contain the whole of her eggs. This gang of
paper-makers, coming one by one to take part in the labour, or perhaps
working without any common agreement, several at a time, at different
points, so far from producing confusion, achieves perfect regularity.
By slow degrees the spacious dome of the summit decreases in diameter;
by degrees it tapers into a cone and ends in a graceful neck.
Individual and almost independent efforts result in an harmonious
whole. Why?
Because these building insects possess an innate geometry, an order of
architecture which is known without being taught and which is constant
in the same group, while varying as between one group and another. Just
as much as the details of the organism, or perhaps even more so, this
propensity to build according to certain determined rules characterizes
the corporations known by the name of species. The Chalicodoma of the
Walls has her earthen tower, the Pelopæus her twisted clay cylinder,
the Agenia her urn, the Anthidium her cotton wallet, the Eumenes her
open-mouthed cupola and the Wasp her paper balloon. And so with the
others: each has her own art.
Our builders contrive and calculate before they set to work. The insect
dispenses with these preliminaries; it knows nothing of the hesitations
of apprenticeship. From the laying of the first stone it is a past
master of its craft. It builds with the same accuracy and the same
unconsciousness as those displayed by the mollusc, which coils its
shell in a scientific spiral; if nothing hinders its aims, it always
achieves a graceful and wisely economical structure. But, when a number
of cells mutually hamper one another, the regulation plan, without
being abandoned, undergoes alterations imposed by lack of space.
Massing leads to irregularity. Here, as with us, liberty makes for
order and constraint for disorder.
We will now open the nest of the balloon-building Wasp. Here is
something that we did not expect. Instead of one envelope there are
two, one enclosed within the other, with a slight interval between.
There would have been even more, three or four of them, had not
impatient hands, eager to bring me the masterpiece, culled it before it
reached perfection. The nest is incomplete, as is proved by the single
story of cells. A perfect Wasps’-nest would contain several stories.
No matter: such as it is, this work shows us that the chilly Wasp was
acquainted with the art of preserving heat before we were. Physics
teaches us the efficacy of a cushion of air, motionless between two
walls, as a preventive against cooling; it recommends the use of double
windows to maintain a mild temperature in our houses in winter. Long
before the days of human science, the little Wasp, that passionate
lover of warmth, knew the secret of multiple envelopes containing
layers of air. With its three or four balloons, fitting within one
another, her nest, hanging in the sun, must turn into a vapour-bath.
These paper containers are merely defensive works; the actual city, for
which all the rest has been built, occupies the top of the dome. It
consists at present of a single layer of hexagonal cells, open below.
Later on, other, similar layers would have been added, descending in
stages and each connected with its predecessor by little papier mâché
columns. The aggregate of these layers, or combs, would supply not far
short of a hundred cells, the lodgings of as many larvæ.
The method of rearing imposes on the Social Wasps rules unknown to the
other builders. These latter store in each cell provisions—honey or
game—apportioned to the grub’s needs. The egg once laid, they close the
cell. The rest does not concern them: the immured larva will find all
around it the wherewithal to nourish itself and to thrive without
outside help. Under these conditions, the irregular grouping of the
cells is of trifling importance; disorder even is admissible, provided
that the whole group be in a place of safety, if need be under the
cover of a protective casing. Richly supplied with provender and
tranquil in its crypt, none of the recluses expects anything from the
outer world.
Among the Social Wasps a very different order of things obtains. Here
the larvæ, from the beginning to the end of their growth, are incapable
of sufficing unto themselves. Like little birds in the nest, they are
fed by mouth; like babies in the cradle, they need constant attention.
The workers, who are celibates expressly appointed to perform household
labours, come and go incessantly, from bed-chamber to bed-chamber; they
awaken the sleepy larvæ, wash them with a lick of the tongue and
disgorge, from mouth to mouth, the ration of the moment. So long as the
larval state continues there is no end to these alimentary kisses
between the nurselings gaping with hunger and the nurses returning from
the fields, their crops swollen with pap.
Nurseries of this kind, which, in the case of various Social Wasps,
number their cradles by the thousand, require ease of inspection,
quickness of attendance and therefore perfect order. Whereas it makes
no difference to the Chalicodomæ, the Eumenes and the Pelopæi whether
their cells be grouped without any great precision, since, once
provisioned and sealed, they will not be visited again, it is important
to the Social Wasps that theirs should be arranged methodically, for
otherwise the enormous household, degenerating into a turbulent mob,
could not possibly be served.
To lodge the mother’s inexhaustible supply of eggs, they have to build
for her, within a limited space, the greatest possible number of cells,
all of a capacity determined by the ultimate size of the larvæ. This
condition exacts a strict economy of the available building-site. No
empty gaps, therefore, which would take up unnecessary room and
moreover compromise the general solidity of the structure.
Nor is this all. The business-man says to himself:
“Time is money.”
The Wasp, no less busy, says to herself:
“Time is paper; paper means a more spacious dwelling, holding a larger
population. Let us not waste our materials. Each partition must serve
two neighbouring apartments.”
How will the insect set about solving its problem? To begin with, it
abandons any circular form. The cylinder, the urn, the cup, the sphere,
the gourd, the cupola and the other little structures of their
customary art cannot be grouped together without leaving gaps; they
supply no party-walls. Only flat surfaces, adjusted according to
certain rules, can give the desired economy of space and material. The
cells therefore will be prisms, of a length calculated by that of the
larvæ.
It remains to decide what form of polygon will serve as the base of
these prisms. First of all, it is evident that this polygon will be
regular, because the capacity of the cells has to be constant. Once the
condition obtains that the grouping must be effected without gaps,
figures that were not regular would be subject to variation and would
give different capacities in one cell and another. Now of the
indefinite number of regular polygons only three can be constructed
continuously, without leaving unoccupied spaces. These three are the
equilateral triangle, the square, and the hexagon. Which are we to
choose?
The one that will approximate most closely to the circumference of a
circle and hence be best adapted to the cylindrical form of the larva;
the one that, with a containing wall of the same extent, will yield the
greatest capacity, a condition essential to the free growth of the
grubs. Of the three regular figures that can be assembled without
vacant intervals, our geometry suggests the hexagon; and it is the
hexagon and none other that is chosen by the geometry of the Wasps. The
cells are hexagonal prisms.
Every high and harmonious achievement finds supersubtle minds that
strive to degrade it. What has not been said on the subject of
hexagonal cells, above all on the subject of the Bee’s, which are
arranged in a double layer and united at the base? Reasons of economy
of both wax and space demand that this base shall be a pyramid formed
of three rhombs with angles of fixed value. Scientific calculations
tell us the value of these angles in degrees, minutes and seconds. The
goniometer subjects the work of the Bee to examination and finds that
the value is precisely calculated to degrees, minutes and seconds. The
insect’s work is in perfect agreement with the nicest speculations of
our own geometry.
There is no room for the glorious problem of the Bee-hive in these
elementary essays. Let us confine ourselves to the Wasps. It has been
said:
“Fill a bottle with dried peas and add a little water. The peas, in
swelling, will become polyhedrons by mutual pressure. Even so with the
Wasps’ cells. The builders work in a crowd. Each builds at her own
will, placing her work in juxtaposition to her neighbours’; and the
reciprocal thrusts produce the hexagon.”
A preposterous explanation, which no one would venture to suggest if
only he had condescended to make use of his eyes. Good people, why not
look into the early stages of the Wasp’s work? This is quite easy in
the case of the Polistes, who builds in the open, on a twig of some
hedge-plant. In the spring, when the Wasps’-nest is founded, the mother
is alone. She is not surrounded by collaborators who, vying with her in
zeal, would place partition against partition. She sets up her first
prism. There is nothing to hamper her, nothing to impose one form upon
her rather than another; and the original cell, free from contact in
every direction, is as perfect an hexagonal prism as the rest will be.
The faultless geometry of the structure asserts itself from the outset.
Look again when the comb of the Polistes or any Social Wasp that you
please is more or less advanced, when numbers of builders are at work
upon it. The cells at the edge, most of them still incomplete, are free
as regards their outer halves. So far as this part is concerned there
is no contact with the preceding row of cells; no limit is imposed; and
yet the hexagonal configuration appears as plainly here as elsewhere.
Let us abandon the theory of mutual pressure: a single glance of the
least discernment contradicts it flatly.
Others, with more scientific, that is to say, less intelligible
ostentation, substitute for the contact of the swollen peas the contact
of spheres which, with their intersections and by virtue of an unseeing
mechanism, lead to the superb structure of the Bees. The theory of an
order emanating from an Intelligence watchful over all things is, to
their thinking, a childish supposition; the riddle of things is
explained by the mere potentialities of chance. To these profound
philosophers, who deny the geometrical Idea Which rules the forms of
things, let us propound the problem of the Snail.
The humble mollusc coils its shell according to the laws of a curve
known as the logarithmic spiral, a transcendental curve compared with
which the hexagon is extremely simple. The study of this line, with its
remarkable properties, has long delighted the meditations of the
geometricians.
How did the Snail take it as a guide for his winding staircase? Did he
arrive at it by means of intersecting spheres, or other combinations of
forms dove-tailed one into the other? The foolish notion is not worth
stopping to consider. With the Snail there is no conflict between
fellow-workers, no interpenetration of similar, adjoining structures.
Quite alone, completely isolated, peacefully and unconsciously he
achieves his transcendental spiral with the aid of glaireous matter
charged with lime.
Did the Snail even invent this cunning curve himself? No, for all the
molluscs with turbinate shells, those which dwell in the sea and those
which live in fresh water or on land, obey the same laws, with
variations of detail as to the conoid on which the typical spiral is
projected. Did the present-day builders accomplish it by gradually
improving on an ancient and less exact curve? No, for the spiral of
abstract science has presided over the scrollwork of their shells ever
since the earliest ages of the globe. The Ceratites, the Ammonites and
other molluscs prior in date to the emergence of our continents coil
their shells in the same fashion as the Planorbes [64] of our books.
The logarithmic spiral of the mollusc is as old as the centuries. It
proceeds from the sovran Geometry which rules the world, attentive
alike to the Wasp’s cell and to the Snail’s spiral.
“God,” says Plato, “is ever the great geometer: Αεὶ ὁ Θεὸς γεομετρεῖ.”
Here truly is the solution of the problem of the Wasps.