The Structure of Comb – Part 2
The Structure of Comb – Part 2
The Bee World – August, 1921 – Pages 73-74
By MISS ANNIE D. BETTS, B.Sc.
What, then, determines the shape of the bee’s cell? It is certainly not economy of wax; for, if the shape of the cell were altered to that theoretically most economical, the saving of wax would only be 1/148th part of the total wax used (1/120th for drone cells). This is insignificant in comparison with the amounts wasted in other ways by the bees. So many theories have been put forward that space forbids a full account of them. One must, however, be mentioned, as it is probably a part of the true explanation.
If we suppose that the bees endeavour to build cylindrical cells with hemispherical bottoms, but start doing this so near to each other that the circles overlap, it is easily seen that cells of the form shown in the diagram will result. Such cells will exactly resemble those investigated by Reaumur and Koenig; a geometrical coincidence which is responsible for some of the confusion that has arisen in connection with this problem.
Some have thought, and no doubt to some extent rightly, that this is how comb is built. As we know from Huber’s account, comb-building is a co-operative business. One bee does a little work, then passes on, and leaves another bee to continue the cell she has been working on. There can thus be no talk of actual interference of circles struck from equidistant centres; nevertheless, each bee is probably well aware when the cell-wall she is working on is thin enough. Consequently, the cells are formed as if geometrical interference were the controlling agency; and all similar cell-walls are finally of the same thickness, because individual bees differ but little in their sense of touch.
The walls, even of the cell-bases, are very thin, and the stresses in the comb must be taken chiefly by the thickened edges of the cells – the lines drawn in the diagram (BEE WORLD, July, 1921, p. 38 – Part 1). Comb may therefore be regarded as a framework of rods; and it can be shown that these rods are so disposed as to distribute the stresses due to the weight of the comb equally to all the internal rods at a given level. The same applies to the external hexagonal network; but the stresses in this are probably different from those in the interior of the comb. In comb turned through a right angle this is not the case; hence the importance of putting in foundation right way up. The distortion due to putting it wrong way up may not be serious; but mathematics tells us that some distortion is bound to occur, and it is better not to take the risk of geting badly-built combs.
As all who have allowed bees to build comb from starters know, a new comb has a characteristic form, being often narrower at the line of attachment than it is lower down. Looked at edgeways, the shape is similar, the comb being drawn out furthest somewhat above the middle, and being wedge-shaped at the edge. Comb built on foundation is also drawn out first in the middle of the sheet. I have never seen any explanation of these facts; so the following may be of some interest.
Comb, as we have seen, consists of a linkage of elastic rods (the internal edges of the cells) enclosed in a skin or network of hexagonal mesh (the cell-mouths). As the internal rods are so disposed as to be stressed to the same extent at the same level – those higher up the comb being more highly stressed than those lower down, on account of the greater weight below them – the interior of the comb may be regarded as mechanically equivalent to the fluid inside a hanging drop. The network of cell-mouths forms a skin enclosing the whole, just as the “skin” of a drop, subject to the forces of surface-tension, encloses the fluid inside it. We should therefore expect to find that naturally-built comb would assume the form of a fluid drop; as it actually does. Its flattened shape (instead of being a “solid of revolution,” like the drop) is due to its bilateral symmetry – to the presence of the midrib. The tendency of comb to be narrower at the line of attachment than lower down is thus explained.
The above is of course only a suggestion, and will have to be worked out mathematically before it can be taken as proved. In the course of this research it may become possible to explain several other puzzling points about comb. Such are: the differences in the cappings of drone and worker cells and of honey cells; the unexpectedly pointed bases, as described by Vogt; and the “pitch,” or upward inclination, of the cells. These last are probably connected; for if we suppose the more pointed base to be produced by the points ACE moving upwards, and the point O moving downwards (see diagram on page 38 of the last number – Part 1), the cell-mouth hexagons will not remain plane unless the edges A’A, C’C, E’E are either shortened or bent (if B’B, D’D, F’F do not alter in length). As it is impossible that A’A, etc., can he compressed without altering B’B, etc. (since they are all of the same extensibility), the cell-edges must bend. This of course assumes that the cells are at first of the form given by Reaumur’s problem; this is a reasonable supposition, not because it gives the minimum surface but because it is the form which will result if the bees try to build cylindrical cells with hemispherical bases, and the parts in contact are flattened by mutual interference.
It is not, naturally, suggested that bees build comb in this way from any instinctive knowledge of the mechanical properties of the structure; they are probably merely influenced by the tendency of the comb to bulge and bend in undesirable places when they add too much to it at any point, compared with the progress of the rest of the comb. The problem is, in fact, one of statics, as Vogt says (though he speaks only of the cell and makes no allusion to the general form of the comb). That the bees are quick to notice any tendency to bulge or break down is certain from their behaviour in repairing damaged combs; may it not well be that it is by this feeling that all is, or is not, well with the equilibrium of the comb that they construct it in the first instance to the pattern of the hanging drop? Their habit of thinning down the edge of old comb before adding to it would be explained by the tendency of the walls of the last row of old cells to distort, if they were of full length, when the weight of the builders was suspended from them at the increased temperature necessary for comb-building.
It has not been possible to give here anything like a full history of the researches on the bee’s cell. The literature alone comprises about 150 references, mostly mathematical; and a fair-sized volume would be required to deal with the matter adequately. It is hoped that the above will, however, give an idea of the problem and its history and will prove of interest, especially in view of the present-day experiments with metal foundation. Should the suggestion above put forward as to the cause of “pitch” be correct, it should be found that comb built on metal foundation is devoid of pitch, if the foundation does not stretch or give at all. I shall supplement these notes in future after examining wax combs built on metal foundation.
By MISS ANNIE D. BETTS, B.Sc.
What, then, determines the shape of the bee’s cell? It is certainly not economy of wax; for, if the shape of the cell were altered to that theoretically most economical, the saving of wax would only be 1/148th part of the total wax used (1/120th for drone cells). This is insignificant in comparison with the amounts wasted in other ways by the bees. So many theories have been put forward that space forbids a full account of them. One must, however, be mentioned, as it is probably a part of the true explanation.
If we suppose that the bees endeavour to build cylindrical cells with hemispherical bottoms, but start doing this so near to each other that the circles overlap, it is easily seen that cells of the form shown in the diagram will result. Such cells will exactly resemble those investigated by Reaumur and Koenig; a geometrical coincidence which is responsible for some of the confusion that has arisen in connection with this problem.
Some have thought, and no doubt to some extent rightly, that this is how comb is built. As we know from Huber’s account, comb-building is a co-operative business. One bee does a little work, then passes on, and leaves another bee to continue the cell she has been working on. There can thus be no talk of actual interference of circles struck from equidistant centres; nevertheless, each bee is probably well aware when the cell-wall she is working on is thin enough. Consequently, the cells are formed as if geometrical interference were the controlling agency; and all similar cell-walls are finally of the same thickness, because individual bees differ but little in their sense of touch.
The walls, even of the cell-bases, are very thin, and the stresses in the comb must be taken chiefly by the thickened edges of the cells – the lines drawn in the diagram (BEE WORLD, July, 1921, p. 38 – Part 1). Comb may therefore be regarded as a framework of rods; and it can be shown that these rods are so disposed as to distribute the stresses due to the weight of the comb equally to all the internal rods at a given level. The same applies to the external hexagonal network; but the stresses in this are probably different from those in the interior of the comb. In comb turned through a right angle this is not the case; hence the importance of putting in foundation right way up. The distortion due to putting it wrong way up may not be serious; but mathematics tells us that some distortion is bound to occur, and it is better not to take the risk of geting badly-built combs.
As all who have allowed bees to build comb from starters know, a new comb has a characteristic form, being often narrower at the line of attachment than it is lower down. Looked at edgeways, the shape is similar, the comb being drawn out furthest somewhat above the middle, and being wedge-shaped at the edge. Comb built on foundation is also drawn out first in the middle of the sheet. I have never seen any explanation of these facts; so the following may be of some interest.
Comb, as we have seen, consists of a linkage of elastic rods (the internal edges of the cells) enclosed in a skin or network of hexagonal mesh (the cell-mouths). As the internal rods are so disposed as to be stressed to the same extent at the same level – those higher up the comb being more highly stressed than those lower down, on account of the greater weight below them – the interior of the comb may be regarded as mechanically equivalent to the fluid inside a hanging drop. The network of cell-mouths forms a skin enclosing the whole, just as the “skin” of a drop, subject to the forces of surface-tension, encloses the fluid inside it. We should therefore expect to find that naturally-built comb would assume the form of a fluid drop; as it actually does. Its flattened shape (instead of being a “solid of revolution,” like the drop) is due to its bilateral symmetry – to the presence of the midrib. The tendency of comb to be narrower at the line of attachment than lower down is thus explained.
The above is of course only a suggestion, and will have to be worked out mathematically before it can be taken as proved. In the course of this research it may become possible to explain several other puzzling points about comb. Such are: the differences in the cappings of drone and worker cells and of honey cells; the unexpectedly pointed bases, as described by Vogt; and the “pitch,” or upward inclination, of the cells. These last are probably connected; for if we suppose the more pointed base to be produced by the points ACE moving upwards, and the point O moving downwards (see diagram on page 38 of the last number – Part 1), the cell-mouth hexagons will not remain plane unless the edges A’A, C’C, E’E are either shortened or bent (if B’B, D’D, F’F do not alter in length). As it is impossible that A’A, etc., can he compressed without altering B’B, etc. (since they are all of the same extensibility), the cell-edges must bend. This of course assumes that the cells are at first of the form given by Reaumur’s problem; this is a reasonable supposition, not because it gives the minimum surface but because it is the form which will result if the bees try to build cylindrical cells with hemispherical bases, and the parts in contact are flattened by mutual interference.
It is not, naturally, suggested that bees build comb in this way from any instinctive knowledge of the mechanical properties of the structure; they are probably merely influenced by the tendency of the comb to bulge and bend in undesirable places when they add too much to it at any point, compared with the progress of the rest of the comb. The problem is, in fact, one of statics, as Vogt says (though he speaks only of the cell and makes no allusion to the general form of the comb). That the bees are quick to notice any tendency to bulge or break down is certain from their behaviour in repairing damaged combs; may it not well be that it is by this feeling that all is, or is not, well with the equilibrium of the comb that they construct it in the first instance to the pattern of the hanging drop? Their habit of thinning down the edge of old comb before adding to it would be explained by the tendency of the walls of the last row of old cells to distort, if they were of full length, when the weight of the builders was suspended from them at the increased temperature necessary for comb-building.
It has not been possible to give here anything like a full history of the researches on the bee’s cell. The literature alone comprises about 150 references, mostly mathematical; and a fair-sized volume would be required to deal with the matter adequately. It is hoped that the above will, however, give an idea of the problem and its history and will prove of interest, especially in view of the present-day experiments with metal foundation. Should the suggestion above put forward as to the cause of “pitch” be correct, it should be found that comb built on metal foundation is devoid of pitch, if the foundation does not stretch or give at all. I shall supplement these notes in future after examining wax combs built on metal foundation.
