Background on the Hexagonal Honeycomb conjecture

In 1994, D. Weaire and R. Phelan improved on Lord Kelvin's candidate for the least-area way to partition space into regions of unit volume. Contrary to popular belief, even the planar question remains open.

-Frank Morgan, Trans. AMS, Vol. 351, No. 5, p1753, 1999.

Let T be a tile of unit area such that the plane may be tiled by congruent copies of it. Steinhaus asks if the perimeter length of T is least when T is a regular hexagon. More generally, if the plane is tiled by bounded tiles, not necessarily congruent, but all of a diameter of at least D0, say, does the regular hexagonal tiling minimize the maximum (perimeter length of T)^2/(area of T) taken over all tiles T in the tiling?

The 3-dimensional analog of this is likely to be challenging: What is the tile T of unit volume and least surface area that permits a tiling of R^3 by congruent copies of T?

-Hallard T. Croft, Kenneth J. Falconer, Richard K. Guy, Unsolved Problems in Geometry (Problem C15). Springer-Verlag, 1991.

Bees are not of a solitary nature, as eagles are, but are like human beings... They have three tasks: food, dwelling, toil; and the food is not the same as the wax, nor the honey, nor the dwelling. Does not the chamber in the comb have six angles, the same number as the bee has feet? The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space.

-Marcus Terentius Varro (116-27 B.C.), On Agriculture, III,XVI. 6. Loeb Classical Library.

In Book V Pappus takes up a topic not mentioned by Euclid, but apparently discussed by the Athenian mathematician Zenodorus.... This topic is the isoperimetric problem: Which plane figure of a given perimeter encloses the largest area? Which solid figure having a given surface area encloses the largest volume? Pappus introduces this problem with one of the most charming essays in the history of mathematics, one that has frequently been excerpted under the title On the Sagacity of Bees. Pappus speaks poetically of the divine mission of the bees to bring from heaven the wonderful nectar known as honey, and says that in keeping with this mission they must make their honeycombs without any cracks through which honey could be lost. Having also a divine sense of symmetry, the bees had to choose among the regular shapes that could fulfill this condition, that is, triangles, squares, and hexagons. They chose the hexagon because a hexagonal prism required the least material to enclose a given volume.

Roger Cooke, The History of Mathematics, A Brief Course, John Wiley and Sons.

Book V [of Pappus] is largely devoted to isoperimetry, or the comparison of the areas of figures having equal bounding perimeters and of volumes of solids having equal bounding areas. This book also contains an interesting passage on bees and the maximum-minimum properties of the cells of their honeycombs.

Howard Eves, An Introduction to the History of Mathematics.

The Pappus Problem

Book V of the Collection was a favorite with later commentators, for it raised the question of the sagacity of bees. Inasmuch as Pappus showed that of two regular polygons having equal perimeters the one with the greater number of sides has the greater area, he concluded that bees demonstrated some degree of mathematical understanding in constructing their cells as hexagonal, rather than square or triangular prisms. The book goes into other problems of isoperimetry, including a demonstration that the circle has a greater area, for given perimeter, than does any regular polygon. Here Pappus seems to have been following closely a work On Isometric Figures written almost half a millenium earlier by Zenodorus (ca. 180 B.C.), some fragments of which were preserved by later commentators.

Carl B. Boyer, A History of Mathematics, Second Edition, John Wiley and Sons.

Zenodorus wrote, at some date between (say 200 B.C. and A.D. 90, a treatise ... On isometric figures. A number of propositions from it are preserved in the commentary of Theon of Alexandria on Book I of Ptolemy's Syntaxis; and they are reproduced in Latin in the third volume of Hultsch's edition of Pappus, for the purpose of comparison with Pappus's own exposition of the same propositions at the beginning of his Book V, where he appears to have followed Zenodorus pretty closely while making some changes in detail.

A History of Greek Mathematics, Sir Thomas Heath, Oxford, 1921.

Book V. Preface on the Sagacity of Bees.

It is characteristic of the great Greek mathematicians that, whenever they were free from the restraint of the technical language of mathematics, as when for instance they had occasion to write a preface, they were able to write in language of the highest literary quality, comparable with that of the philosophers, historians, and poets. We have only to recall the introductions to Archimedes's treatises and the prefaces to the different Books of Apollonius's Conics. Heron, though severely practical, is not exception when he has any general explanation, historical or other, to give. We have now to note a like case in Pappus, namely the preface to Book V of the Collection. The editor, Hultsch, draws attention to the elegance and purity of the language and the careful writing; the latter is illustrated by the studied avoidance of hiatus. The subject is one which a writer of taste and imagination would naturally find attractive, namely the practical intelligence shown by bees in selecting the hexagonal form for the cells in the honeycomb. Pappus does not disappoint us; the passage is as attractive as the subject, and deserves to be reproduced.

`It is of course to men that God has given the best and the most perfect notion of wisdom in general and of mathematical science in particular, but a partial share in these things he allotted to some of the unreasoning animals as well. To men, as being endowed with reason, he vouchsafed that they should do everything in the light of reason and demonstration, but to the other animals, while denying them reason, he granted that each of them should, by virtue of a certain natural instinct, obtain just so much as is needful to support life. This instinct may be observed to exist in the very many other species of living creatures, but most of all in bees. In the first place their orderliness and their submission to the queens who rule in their state are truly admirable, but much more admirable still their emulation, the cleanliness they observe in the gathering of honey, and the forethought and housewifely care they devote to its custody. Presumably because they know themselves to be entrusted with the task of bringing from the gods to the accomplished portion of mankind a share of ambrosia in this form, they do not think it proper to pour it carelessly on ground or wood or any other ugly and irregular material; but, first collecting the sweets of the most beautiful flowers which grow on earth, they make from them, for the reception of the honey, the vessels which we call honeycombs, (with cells) all equal, similar and contiguous to one another, and hexagonal in form. And that they have contrived this by virtue of a certain geometrical forethought we may infer in this way. They would necessarily think that the figures must be such as to be contiguous to one another, that is to say, to have their sides common, in order that no foreign material could enter the interstices between them and so defile the purity of their produce. Now only three rectilineal figures would satisfy the condition, I mean regular figures which are equilateral and equiangular; for the bees would have none of the figures which are not uniform... There being then three figures capable by themselves of exactly filling up the space about the same point, the bees by reason of their instinctive wisdom chose for the construction of the honeycomb the figure which has the most angles because they conceived that it would contain more honey than either of the two others.

`Bees then, know just this face which is of service to themselves, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material used in constructing the different figures. We, however, claiming as we do a greater share in wisdom than bees, will investigate a problem of still wider extent, namely that, of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always greater, and the greatest plane figure of all those which have a perimeter equal to that of the polygon is the circle.'

Sir Thomas Heath, A History of Greek Mathematics, Oxford, 1921.

Chapter I The beginnings of geometry

The subject of geometry, whether we consider it as a science or an art, has a very long history.... What is the earliest example of purposive geometrical construction.... If a pigeon fly straight rather than on a curve, once it has decided the direction of its home, we are safe in saying that it has intentionally taken the shortest path and applaud its sagacity, without inquiring as to whether the real reason was not that it had an impulse to go straight, and none to bend....

Another example in Nature where great credit has been given for geometrical sagacity is found in the cell-structure of the honey bee. Such a cell is a prism whose section is approximately a regular hexagon, while the ends are three-faced `steeples'. The cleverness of the creature in constructing a habitation of hexagonal section excited the admiration of Pappus, some sixteen hundred years ago:

[Heath's translation of Pappus is quoted here.]

But the bee's skill in solving problems in maxima and minima does not end here. It was found on examination that the angle formed by the faces of the steeple with the axis was such as to make the total surface for a given volume a minimum, a result experimentally determined by MacLaurin. What clever bees!

But were the really so clever? Let us look at the matter more closely....

Julian Coolidge, A History of Geometrical Methods, Dover.

The Sagacity of the Bees in making their Cells of an hexagonal Form, has been admired of old; and that Figure has been taken notice of, as the best they could have pitched for their Purposes: But a yet more surprising Instance of the Geometry of these little Insects is seen in the Form of the Bases of those Cells, discovered in the late accurate Observations of Monsieur Maraldi and Monsieur de Reaumur, who have found those Bases to be of the Pyramidal Figure, that requires the least Wax for containing the same Quantity of Honey, and which has at the same time a very remarkable Regularity and Beauty, connected of Necessity with its Frugality.

Colin Mac Laurin, June 30, 1743.

Much has been written on the this question of the geometry of the honeycomb. The bee's strange social habits and geometric talents could not fail to attract the attention and excite the admiration of their human observers and exploiters. ``My house,'' says the bee in the Arabian Nights, ``is constructed according to the laws of a most severe architecture; and Euclid himself could learn from studying the geometry of my cells.''

Darwin ... spoke of the bees' architecture as ``the most wonderful of known instincts'' and adds: ``Beyond this stage of perfection in architecture natural section (which has replace divine guidance!) could not lead; for the comb of the hive-bee, as far as we can see, is absolutely perfect in economizing labor and wax.'' Weyl, Symmetry, 1952.

The most famous of all hexagonal conformations, and one of the most beautiful, is the bee's cell.

D'Arcy Wentworth Thompson, On Growth and Form, Vol 2, page 525, second edition, 1952.

What Jeremy Taylor called ``the discipline of bees and the rare fabric of honeycombs'' must have attracted the attention and excited the admiration of mathematicians from time immemorial. ``Ma maison est construite,'' says the bee in the Arabian Nights, ``selon les lois d'une sévère architecture; et Euclidos lui-même s'instruirait en admirant la géométrie de ses avéoles.'' Ausonius speaks of the geometrica forma favorum, and Pliny tells of men who gave a lifetime to its study.

Pappus the Alexandrine has left us an account of its hexagonal plan, and drew from it the conclusion that the bees were endowed with ``a certain geometrical forethought''. ``There being, then, three figures which of themselves can fill up the space round a point, viz. the triangle, the square and the hexagon, the bees have wisely selected for their structure that which contains the most angles, suspecting indeed that it could hold more honey than either of the other two.'' [footnote: This was according to the ``theorem of Zenodorus.'' The use by Pappus of ``economy'' as a guiding principle is remarkable. For it means that, like Hero with his mirrors, he had a pretty clear adumbration of that principle of minima which culminated in the principle of least action, which guided eighteenth-century physics, was generalised (after Fermat) by Lagrange, inspired Hamilton and Maxwell, and reappears in the latest developments of wave-mechanics.]

Thompson, On Growth and Form

Thompson refers to ``countless papers on the bee's cells'' and cites a long list from the 19th century on page 544 of Vol 2 of On Growth and Form.

Aristote déjà avait remarqué que les abeilles commencent le réseau des cellules par le haut de la ruche, et descendent en juxtaposant des rangs d'hexagones. Après lui, il n'est guère d'écrivains traitant de biologie ou d'agriculture qui n'en aient parlé. L'agronome Varron hésitait entre deux explications: ou bien six côtés parce que l'abeille a six pattes, explication naïve qui se retrouve chez Pline, ou bien occupation maximale de l'espace, idée que Kepler développera et que les zoologistes modernes semblent admettre. Les autres auteurs de l'Antiquité se sont bornés à dire que l'abeille sait la géométrie.

Robert Halleux, Introduction to L'Étrenne ou la neige sexangulaire, 1975.