The Economy and Symmetry of the Honey-Bees' Cells. [From Runkle's Mathematical Monthly 2.9 (June 1860): 304-319.] ¹⁰

By Chauncey Wright, Nautical Almanac Office, Cambridge, Mass.

The economical characteristics of the honey-cell have claimed so great attention from mathematicians, that other and even more prominent properties of its structure, though not unnoticed, have received too little attention. Psychologists, accordingly, following the testimony of mathematicians, have treated of the instinct of bees as if economy were not simply the most important, but even the only useful and noticeable feature of the bees’ architecture. In reconsidering therefore the geometrical properties of the hive-cell we have chiefly in view the inferences which may be drawn from them in regard to the nature and powers of the bee’s instinct.

In this review we shall first recount the reasoning a priori by which the structure of the hive-cell is deduced from considerations of rational economy, and then, in the second place, deduce the same results from considerations of symmetry applied to those modifications of the simple nest which are required by simple economy.

It is necessary here to premise a division of economy into two species, which we may denominate rational and sensible, or the

economy of forethought and simple economy; that which forestalls waste, and that which remedies waste or simply saves. The importance of this distinction will be apparent when, in the third place, we consider the two previous lines of argument in their relation to the faculties of instinct.

1. It is a proposition of elementary geometry that of all polygons or rectilinear plane figures with a given number of sides and a given perimeter, that figure contains the greatest area which is regular, or both equilateral and equiangular; and hence, that regular prisms have the greatest solid contents for a given convex surface. Again it is proved that a given perimeter encloses the greatest area in that regular polygon which has the greatest number of sides, so that among plane figures the circle has the greatest area for a given perimeter; and hence, also, the cylinder has the greatest solid contents among prismatic bodies for a given convex surface.

If now we seek among all regular polygons, or prisms, for those which are capable of dividing space into equal and similar parts without interstices!, we readily perceive that the angles of such a figure must be aliquot parts of the circle or of four right angles. All the angles of any such figure are equal to twice as many right angles as the figure has sides minus four right angles, or if a be the number of sides, the sum of all the angles is (2 n - 4) right angles; hence each angle of a regular figure must bear the ratio 2n -4/n to a right angle, and as 4 right angles must be divisible by each of these angles, we have 4n/2n-4, or 2n/n-2 = integer. But this number is equal to 2 + 4/n-2; hence (n - 2) must be a divisor of 4, that is, either 1, 2, or 4, and n must therefore be either 3, 4, or 6; so that triangular, quadrilateral, and hexagonal regular polygons and prisms are the only ones which can collectively fill space without interstices. Now of these the one which has the greatest

number of sides is the most economical. Hence the partition into regular hexagonal prisms is the most economical division of space in respect to two dimensions; but the third dimension is left undetermined.

If we suppose a hollow space, bounded by the surface of a regular hexagonal prism, to be open at one end, but closed at the other, so as to form a cell, it is clear that the base of this cell is most economized, if, like the sides, it be made also a boundary to another cell, or partially to several other cells. Hence, if two series of hexagonal cells, opening in opposite directions, have common bases, they present another feature of economy which is also exhibited in the form of the honeycomb. To this extent the bees’ economy was known to the ancients.

There are two forms in which the bases of the cells might be fashioned so as to fit them as bases also of the opposite cells in the comb. They might be either plane hexagons perpendicular to the sides of the cells, or be composed of three rhombs inclined to the sides of the cells, and forming a solid angle in the central line or axis of the cell. If in the first form all the bases be in the same plane the relative positions of cells on opposite sides of the comb are immaterial to economy. But as each angle of the regular hexagonal prism is 4/3 of a right angle, or 1/3 of the circle, three regular hexagonal prisms have one common comer or line of contact, and this common comer must, in the second form of the bases, be in the same line with the axis of an opposite cell, the base of which is therefore composed of three thirds of the three contiguous bases.

The dimensions of these rhombic segments of the bases vary with their inclinations to the sides and axes of the cells. One diagonal however is the same in position and length for all inclinations of the rhombs, while the other diagonal will vary with its inclination to the axis of the cell from the length of the shorter diagonal in the rhombs of a plane hexagonal base to any length whatever. This increase in the length of the variable diagonal increases the area of the rhombs, while the sides of the cells are more and more cut off and their area diminished as the rhombs become larger and more obliquely inclined.

In passing from the form of the plane hexagonal base, the sides at first diminish more rapidly than the rhombs increase, while the solid contents of the cell is the same for all the inclinations and corresponding forms of the rhombic segments of its base. It becomes a problem, therefore, of economy to determine that inclination and form of the rhombs which will give the least value to the sum of the areas in the sides and bases, or to the whole surface of the cell.

Instead of giving here the ordinary algebraic solution of this problem, which gives the same form to the rhombs as that determined by Maraldi's measurements of the honey-cell, we shall proceed . upon the second line of argument to deduce the same form from considerations of symmetry, and we will then prove by geometrical reasoning that this form is a minimum. We may remark here, however, that Maraldi determined by measurement, not merely the angles of the rhombs, but also a symmetry in the structure, which, as some geometricians have supposed, might have enabled him by calculation to give these angles, 109° 28' and 70° 32', with so great precision. The symmetry which Maraldi observed was the equality of the angles that the planes of the rhombs make with each other to the angle 120° which the sides of the cell make with each other, and also with the rhombs. This symmetry depends

directly on the regularity of all the solid angles in the cell, which Maraldi also pointed out. Indeed, if we regard only the angles which the planes make with each other, there is but one angle, 120°, in the whole structure. This one symmetry affords a perfect rule to the insect architect.

If we wish further to determine the angles which the lines make with each other, it is not necessary to make any direct measurements, but only to calculate the angles at the vertex of a triangular pyramid of which the sides make with each other the angle 120°. These angles are incommensurate in the circle, but in the triangle they are as simple as the angles 120° and 60°; for while the cosines of the latter are ± 1/2 the former angles and their supplements (approximately 109° and 71°) have cosines exactly ± 1/3; and as the trigonometrical measurement of angles is the most easy and natural, these angles are as easily constructed as the others.

2. But why, it may be asked, should the bees’ instinct prefer a structure, in which all the angles of the planes are 120°, to any other, unless this structure be also the most economical? That it is the most symmetrical structure of cells that can be imagined will be granted, in spite of its seeming complexity; but of what advantage is elegant symmetry to the bee, unless it also economizes labor and material? And what therefore could have fashioned the instinct of the bee except a supersensible principle of rational foresight, superior to mere sensible perception? In considering these questions let us look at the hive-cell from another point of view. Having built it up a priori from general principles, let us see in what relations it stands a posteriori to other cells.

We learn from naturalists that the cell is a species of nest or open cocoon, and that cocoons and nests are of two kinds, excavations and structures; but both kinds have these general characteristics of form. If closed, they are either spheres or cylindrical

figures with, hemispherical ends; and if open, they are either hemispheres or cylindrical figures open at one end and terminated at the other by hemispheres. The latter form of cocoon, nest, or cell is then the natural type of the honey-cell. When finished the cells resemble a series of excavations in wax, though in fact they are structures. That a single excavation in sand or wood—a cylindrical pit terminated spherically—has even less surface for the same contents than the honey-cell, has never excited remark on account perhaps of other and more obvious utilities in such a simple symmetrical form.

If a series of such excavations be made as closely together as possible, there would result an arrangement like that of a pile of equal cylinders, each pit surrounded by six other pits, and though the surface of each cell would be enlarged in a greater proportion than the enclosed space by changing the figures of the excavations, still space would be gained, and, in case of a structure, material would be saved, by converting the cylindrical cells into regular hexagonal prisms, (as may be seen in Fig. 2); and this too by simple saving, or by the economy of afterthought. Again, if equal spherical excavations be made as closely together as possible there would result an arrangement like that of a pile of equal spheres (Fig. 2), each cavity surrounded by six others in the same layer, and by three others in each of the contiguous layers,—in all, by twelve spherical cavities. The walls that would be left by removing

the superfluous material of the interspaces or comers would bound regular hexagonal prisms, terminated at both ends by pyramidal bases (Figs. 4,5), each of which would be composed of three rhombs of the same form as the rhombs of the honey-cell. If, therefore, we were to put into the cells of the honey-comb little spheres which would just fit them, and should press them to the bottoms of the cells on both sides of the comb, the spheres would touch the middle points of the rhombs and the middle lines of the sides, so that the sides and bases of the cells would be tangent planes to those points at which the spheres would touch each other but for the thickness of the intervening walls.

To prove this, let us consider two spheres in contact in two contiguous layers of a pile. It will be seen from Fig. 2 that the projections of the centres of two such spheres upon the ground plane are distant from each other by the length of a side of the circumscribed hexagon. If, therefore, in Fig. 3 we draw two vertical parallel lines, at the distance O M apart, equal to the side of the hexagon circumscribed around the circle O, then O and O′ upon these two lines, at a distance apart of twice the radius of the circle, will represent the positions of the centres of the two spheres in contiguous layers. A, bisecting O O′, is the point of contact of the two spheres, and B C, perpendicular to O O′, is tangent to the two spheres, and is the shorter axis of the tangent rhomb. But O O′, equal to P R or to twice the radius of the spheres,

is equal to the longer axis of the rhomb. Hence by drawing C O′ and B O we have a figure, G O B O′, of the same form as the tangent rhomb. If this figure were tinned around its shorter axis, B C, to a position perpendicular to the plane of the diagram, it would coincide with the tangent rhomb. Its dimensions can be readily computed.

If r denote the radius of the spheres or circles, then the side of the circumscribed hexagon is obviously r √4/3, and as O O′ = 2r we find O′ M =r √4/3. As the ratio of the axes of the rhomb is equal to the ratio of O′ M to O M or to √8/3 ÷ √4/2 = √2, we find B C = 2r/√2 = r √2. Hence the sides of the rhomb are equal to r √3/2. B M is therefore equal to r(√8/3- √3/2) = r √1/6 and the ratio of B M to B O is √1/6 ÷ √3/2 = 1/3, which is therefore the cosine of the angles of the rhomb. Hence these angles are the same as the angles in the rhombs of the honey-celL This may be shown still simpler, without computation, by observing after Maraldi, that the solid angles of the honey-cell are all regular, that is, composed of equal plane angles. The triangular ones, which are formed by the intersections of the three rhombs and by the intersection of each rhomb with two sides of the cell, have the common plane angle 109° 28′ nearly, while the quadrilateral solid angles formed by the intersection of two rhombs and two sides have the common plane angle 70° 32′ nearly (see Figs. 4, 5). Now as all the spheres which can be drawn tangent to all the sides of a regular solid angle must touch them in lines that bisect them, it follows that a sphere touching all the sides and rhombs of the honey-cell must touch the rhombs in the intersections of these bisecting lines or in their middle points, and the sides in their middle lines. In fine, the form of the honey-cell is that which would be obtained by placing equal spheres in two layers, as in Fig. 2, and drawing tangent planes through their points of contact, and terminating these planes in

their mutual intersections. These planes may be regarded, without reference to the contact of spheres, as planes midway between equidistant points in two parallel series. They would therefore pass through the intersections of such equal spheres as have radii greater than half the distance between their centres. Fig. 3 also represents portions of two sides of the cell turned up into the plane of the rhomb around the comers O B and O′ B. It also shows that the angle made by the plane of the rhomb, or by its shorter axis with the axis of the cell, is half the larger angle of the rhomb.

We have thus determined the form of the honey-cell from principles of symmetry alone. That this form is the most economical is easily shown by supposing it to vary, and by determining the changes of area in the sides and bases.

Let us suppose, as before, the rhomb C O B O' to be turned upon the axis B C by a right angle, so that the corners O and O' may fall in direction upon A, and let the sides that are attached be turned downward into their positions as vertical planes tangent to the sphere around O. In turning the rhomb now around its longer axis, the comers C and B must remain in the vertical lines C O and B O', so that the shorter axis is diminished if B is raised, and increased if B is depressed. Let i be the amount by which B is raised or depressed; then the projection of i upon the shorter axis B C of the rhomb is half of the amount by which this axis is made shorter or longer. The projection of i upon B C reduces it in the proportion of the semi-minor axis to the side of the rhomb or by the ratio r √½ ÷ r √3/2 hence the projection is i × √⅓. Half of this multiplied by the semi-major axis r is the change of area on each edge of the rhomb; hence the whole rhomb is decreased or increased by 2 r . i . √⅓. But ½ i multiplied by the breadth of the side of the cell is the corresponding change in each of the sides attached to the rhomb, and hence both sides together are increased or decreased by

i · r √4/3 = 2 r.i.√1/3. The sides gain as much therefore as the rhombs lose, or lose as much as the rhombs gain, by an infinitesimal change in their positions, and form; and the whole surface is therefore either a maximum or a minimum, while the contents of the cell is. unchanged by such a change of form.

Again, the infinitesimal triangles on either side of the line B C, resting on the bases i, are the amounts which the sides of the cell are made to gain or lose, while the smaller infinitesimal triangles on either side of B C, and resting on the projections of i, are the amounts which the edges of the rhombs lose or gain. These smaller triangles are in vanishing equal to half the larger ones, since the infinitely smaller right triangles resting upon i are similar to O' A B, the hypothenuse of which, O' B, homologous to i, is bisected by the line D E, and to this line the sides of the smaller infinitesimal triangles become parallel. If i be finite, the decrease of the minor axis of the rhomb will be less than twice the projection of i upon B C, and its increase will be more than twice this quantity; hence the rhomb will lose less than the sides gain, or gain more than the sides lose, by a finite change in the positions and form of the rhombs. The whole surface is therefore a minimum. Thus the symmetrical form of the honey-cell is seen to be also a minimum form.

This economy however has no reference to the depth of the cell, so that geometricians have conceived of a further limitation of the cell’s form, which is not observed by the bee. Having determined the most economical arrangement of the sides and bases, we may further limit the length and breadth of the sides to that ratio, which gives what has been called and determined as the minimum minimorum form of the cell. If A denote the whole area of the cell’s surface, C the whole solid contents, and l the length of the longer edges of the cell, or of those which terminate in the longer axes of the rhombs, (Figs. 4, 5,) and if, as before, we denote by r the radius of

the inscribed sphere or cylinder, then from the dimensions already computed we find
A = 2 r2 √2 + 4 lr √3, C = 2 r2 l√3.

If one of these be supposed constant and the other a minimum, their derivatives must both be equal to zero. Hence by differentiation and reduction we have the equations
r√2 + r Dr l√3 = 0, 2l + r Dr l =0,

from which, eliminating Dr l, we obtain l r√2/3. Hence, l in the minimum minimorum cell is half the distance O′ M between the two planes on which the centres of the contiguous inscribed spheres are arranged. This form of the honey-cell is therefore what the bee would fashion if, instead of its deep nests, it should construct hemispherical nests like the bird, and then convert them into polyhedral figures with sides tangent to the contiguous hemispheres on both sides of the comb. Two such cells would exactly enclose a sphere with tangent planes, as in Figs. 4, 5. Of all figures, therefore, which can divide space into equal and similar parts, without interstices, the polyhedrons of Figs. 4, 5 have the least surface for a given contents.

These figures are such as would be formed by drawing tangent planes through the twelve points of contact on a sphere surrounded by twelve equal spheres. The two forms arise from the two ways in which the three spheres, in each of the outside layers, may be arranged. It may be seen in Fig. 2 that the three superposed spheres occupy alternate interstices; so that there are two ways in which they may be placed in contact with the central one. If the three spheres, which are in contact with the central one from below, be opposite

the three superposed ones, we have the polyhedron of Fig. 5; if alternately disposed, we have that of Fig. 4, the sides of which are all equal and similar rhombs, like the rhombs of the honey-cell. Fig. 5 may be produced by turning the lower half of Fig. 4 one sixth of its circumference around its vertical axis.

Geometricians have also criticised the economy of the bee by computing the difference between the surface of cells with plane bases and the surface of honey-cells having the same contents; and by showing that this difference is only about 1/50 part of the whole surface of a completed cell; so that the bee is enabled by the pyramidal form of the base to make 51 completed cells, instead of 50, out of the same material,—a gain hardly worth so great precision in construction.

3. But this criticism would be just only on the suppositions, 1st, that a plane base could be more readily made than a pyramidal one, and 2dly, that the instinct of the bee is determined to a structure the most economical a priori and on the whole. These suppositions are not however necessary, nor are they supported by facts.

It is true, indeed, as we have seen, that tire form of the honey-cell requires less material than any similar one for the construction of a double series of cells like the honeycomb. But we have also seen that simple nests are bounded by still less surface than the honey-cell, in proportion to their contents, and that the deviation of the single honey-cell from the cylindrical nest with a hemispherical base does in fact increase the surface in a greater proportion than it increases the Enclosed space,—with a gain however of material when the cells are contiguous. As their bases are boundaries between

between cells on opposite sides of the comb, the pyramidal form would be possible as a modification of the hemisphere only in case of a close symmetrical arrangement of the opposite cells. But for such an arrangement,—of two series of spherical bases in closest contact, — a modification to the pyramidal form of the honey-cell is not merely the most economical, but also the simplest, and one perfectly analogous to the hexagonal modification of the sides.

That the cell ought to be regarded as such a modification of the simple nest by ample economy will be evident when we consider the mode of its construction. If the cells were excavated nests made in a costly material, they would by the closest arrangement and the removal of the interstitial material, receive through the agency of simple economy the hexagonal form of the honey-cell; but the bases of opposite cells could only by accident meet as they do in the honeycomb, and hence they would either retain, without reference to each, other, the hemispherical form, or be flattened to plane hexagonal bases. But the hive-bee does in fact build up its nests from their bases which are rudely fashioned at first on the edges of the comb in the form of little cavities at equal distances. On one side of the comb these cavities form by their depressions the interstitial elevations between the cavities of the other side; and hence the symmetrical arrangement of the two series of cells, and hence also the pyramidal form of the bases and their economy.

It has been observed that mechanical pressure alone is capable of changing the elongated cells of cellular tissues—as in the fruits and piths of some plants—into the polyhedral form of the honey-cell. This however is not the way in which the honey-cell is formed, for the bees fashion the corners of their cells as they build, by continual trimming and saving.

Again, the bee builds with great precision; indeed, the exactness of the structure is more surprising than its economy. But precision

is not required by economy at those limits of form which determine maxima and minima values; for these values are determined by the condition that they shall vary least by slight changes in the forms on which they depend. Symmetry, on the contrary, requires absolute precision and affords the means of effecting it. Thus, when the bee’s point of view is just over each one of the middle points of the rhombs it ought to be at the same distance, from all the nine places of the cell, and just opposite similar points of view in the nine contiguous cells. This symmetry affords a working guide to the bee as perfect as the regularity of the solid angles or the equality of all the angles made by the planes with each other.

It is true that these symmetries also determine the economy by which the honey-cell is characterized; so also does the still simpler symmetry of the cylindrical and hemispherical nest determine a still greater economy of work and surface in the architecture of innumerable other nest-building animals. But have psychologists, therefore, thought it necessary to regard the instincts of these animals as determined by supersensible properties of form, instead of that facility of construction and simple saving which are apparent to the senses?

Many deviations not only from the strictest economy, but also from symmetry, are observed in the honeycomb, and are required by the various sizes and uses of the cells.

In the normal form of the comb the triangular comers are formed by the meeting of four cells or by the intersection of six planes, three belonging to the bases and three to the sides; while the quadrilateral comers are formed by the meeting of six cells, or the intersection of twelve planes, six belonging to the bases and six to the sides. Now the bees very frequently in finishing their cells do not make these twelve planes meet in a common point, but connect them by a thirteenth very small plane, forming a tenth partition or

a fourth basal segment to two such cells on opposite sides of the comb as would otherwise only touch each other by a common comer. This supplementary basal segment is however perpendicular to the line which joins the symmetrical centres of the two cells, and is therefore easily included in the symmetries of the cell by the following considerations.

We have seen that if the lines joining equidistant points in two parallel planes be bisected by planes perpendicular to them, these planes terminating in their mutual intersections will form the corners of the honey-comb. Such lines of juncture form the edges of the regular tetrahedrons and octohedrons into which the space between the two planes may be divided; each tetrahedron surrounded by three octohedrons, and each octohedron by six tetrahedrons. The six planes that bisect the six edges of each tetrahedron meet in the centre of the figure and form four contiguous triangular comers. The twelve planes that bisect the twelve edges of each octohedron form in the centre of the figure six contiguous quadrilateral comers; and if in addition to these planes three be drawn midway between each pair of diagonal comers in the octohedron, they will also pass through the centre of the figure, but would be of no service as partition walls between these comers in a perfectly regular figure. If, however, the points in the parallel planes be not exactly equidistant, and the included figure vary from a regular octohedron, the twelve planes cannot at the same time bisect the edges of the figure and meet in the centre; and the bee accordingly, forced by this necessity, instead of making these planes meet in a common point, chooses to join them by another small one midway between two diagonal comers of the figure. Thus the bees separate the central points of their nests by planes midway between the centres of contiguous nests on the same side and on opposite sides of the comb, and also in irregular structures, by smaller planes

between the centres of such nests on opposite sides of the comb, as would otherwise only touch each other in a common quadrilateral comer. This way of remedying irregularities of structure shows conclusively that the bee’s instinct is chiefly governed by symmetries of partition instead of those that belong to any particular figure.

Though it is to be presumed from what we have seen, that special facilities and economies of labor and material govern all abnormal forms, yet habit, undoubtedly, must greatly modify them from special adaptations to general conformities of structure.

The symmetries which we have studied are obviously not the same to the bee and to the geometrician, for what the latter comprehends abstractly as symmetry of form, is only perceptible to the former concretely, as facility of construction. Like the two kinds of .economy, the one is rational, the other sensible.

An unreflective and unforeseeing economy, which, without reference to an end, simply saves, through sensuous preference, what the conditions of life render useful and costly to the race, characterizes the whole animal kingdom. For even those animals which do not store food or build structures for themselves or their offspring, still select through sensuous preference the food that is nutritious and appropriate to them, without foreseeing its use. An application of such a simple saving to the typical structure of the nest results in the honey-cell. The bee’s instinct ought not therefore to be regarded as an exception to animal instincts in general; much less ought animal instincts in general to be so interpreted as to include a misconception of the bee’s instinct.

A natural choice from simple feelings and from heritable sensuous associations — empirical in form, whatever their origin—is all that is required to account for the most perfect work of instinct.