Mathematics Of A Honeycomb (1964)

Everyone knows some things about the honeybee. Her industrious buzzing from flower to flower seeking nectar. The delicious product she makes from that nectar. and the potency of her last resort defense. Such things as are commonly known about the bee do not suggest a beehive as a good place for a lesson in mathematics. But here, as in all of nature, understanding what is happening involves mathematics. For mathematics is the language of science. This piece of honeycomb, a little over an ounce, represents over 20,000 B miles for collection of nectar, the raw material, plus over 110,000 B hours spent transforming the nectar into wax, to which must be added more than 18,000 B hours for fashioning the wax into comb according to a precise pattern. Since honeycomb is so expensive to construct, it's obvious that economy in the use of wax is very important to the welfare of the hive. Is there a geometric shape which is more economical to build than any other? We have made models in which each has the same volume and the same height. Which of these requires the least amount of material to build? that is which has the smallest lateral surface area. Since the height is the same, the one with the shortest perimeter is the best. More sides mean less perimeter. Thinking of the cylinder as having an infinite number of sides shows that it uses the least material. But cylinders are only economical if they stand alone. placed together. They leave large gaps between them. Since no walls can be shared, they are wasteful of both space and material. Wherever walls lie side by side, they can be shared by two cells. That is, one wall can do the work of two. Thus, octagons in which four of eight sides can be shared save 25% on material. Triangles by sharing all their sides and squares save 50% on material. Therefore, when we consider the savings possible through the sharing of sides, the material needed to construct cylinders does not change. But the other figures require less material for they share sides. Of all the figures, the hexagon uses the least material. This is the shape the bees use. This discovery of the economy of honeycomb was made over 16 centuries ago by an Alexandrian geometer, Papus. What he and others have learned about honeycomb has been applied to build the largest grain elevator in the world. The hexagon provided the most economical, the strongest design. Such study of living things is today called bionics. Much effort is being expended to learn from a wide variety of creatures and results are coming in. Study of the eye of a beetle has resulted in a device installed on planes which measures their ground speed more accurately than had been possible before. Study of the common housefly has given us a new extremely rugged gyroscope. Indeed, there seems to be no area in which men cannot learn from the abilities and organs which God has given to his creatures. And in practically every instance, mathematical calculations are involved in understanding the animals and in applying what we learn. There is much more we can learn from honeycomb. A structural engineer can show us other advantages of the hexagon among all the patterns the bees might have used. One of his tools is the polaroscope. When certain transparent plastics are placed in a polaroscope and a stress applied, colored patterns appear. The series of rainbow changes are most numerous where the stress is most concentrated as at the corner of the notch. the weakest point on the bar. The stress patterns from various geometric shapes are most revealing. What if bees had built square cells? One of the drawbacks of square cells is that a load along a partition is not readily transferred to adjacent partitions. In triangular cells, a load at any point is distributed to several members. However, there is a major drawback. The horizontal members are loaded in compression. This is the wrong way to load a thin plate which readily buckles in hexagonal cells. However, the load is distributed as a tension. This makes maximum use of the strength of the thin wax walls. The pattern is so strong that less material is needed in the walls to support the load. Bees take advantage of this and make the walls of the cells less than 3000 of an inch thick, thinner than a sheet of paper. It's amazing, isn't it? Bees build honeycomb just as if they were graduates from the best schools of engineering using the shape which is not only the most economical but also the strongest. The understanding of the polaroscope pattern by an engineer requires a great deal of mathematics. In fact, to go into a thorough study of the structure of honeycomb involves mathematics so complex as to require a computer. Let's look further at the way in which mathematics, some of it quite advanced, has enabled the scientist to understand the structure of the honeycomb. Some two and a half centuries ago, a French astronomer Moraldi became fascinated with bees. One thing he noted about the double rank of cells in honeycomb is that their bottoms are not flat but convex formed of three rabi. Obviously, the obtuse angles where bottom meets wall make it easier for the bees to construct the cells and to keep them clean. But Moralti noted that there was a remarkable constancy to the angles of the rambi. They measured about 70°. He suggested that the bees had used this angle for the sake of simplicity. For if the acute angles of the rai are 70° 32 minutes, the angles of the side trapezoids are identical. Ryamure, a noted French scientist, suspected that there was a further reason for the special shape. He wrote to a number of mathematicians asking what angle would give the most economical trihedral base on a hexagonal prism. Only one Kernig came up with an answer. To get his answer, he first proved that the volume of all hexagonal prisms with trihedral bases is the same provided that the height h and the side a remain constant. This is easy to see. If a piece is cut off one place and put back on at another, the volume does not change. But as this distance changes the shape of the prism and the area of its surface change. If this distance is called x, this is the formula for the surface area. It consists of the areas of the six rectangular sides less the triangular pieces plus the areas of the three rhombic ends which can be calculated by means of the Pythagorean theorem. Substituting in this formula, calculating values for x= 0, x= 110th and so on. Plotting these values x = 4/10, 6/10 and so on. And then connecting these points gives us a graph which shows that the area is at a minimum when x lies between 3/10 and 4/10. In other words, the prism with the flat base and the prism with the very pointed base have more surface area and would require more material to build than the intermediate one. Unfortunately, no amount of graphing or even brute force calculation will yield a mathematically precise answer. But there is an elegant shortcut to an exact answer. Some 300 years ago, Newton invented the calculus. About 75 years later, when few men understood the method, Kunig manipulated the formula for the area according to the precise rules of the calculus. The mathematician calls it differentiating with respect to x and equating to zero to minimize. This process yields the precise point where the surface is at a minimum. With complete generality, this occurs when x= a, the length of a side, times the square root of 2 / 4. The cell then has the shape that B uses with the angles of the rami that moral gave. Someone may wonder whether this is so economical. Isn't it necessary to waste wax to build two pointed ends? Bees have the answer to this. They build the cells so that the bottoms are offset. Each cell fits into the pocket formed by three cells on the opposite side. Modern science with its high-speed computers and intricate formulas for stress analysis has not been able to improve on the honeycomb. It has merely confirmed it as the ideal structural shape. But man could not understand the perfection of this pattern which the creator had given to the honeybee until after he understood the mathematics involved. For as Galileo noted, that vast book which stands forever open before our eyes, I mean the universe, cannot be read until we have learned the language. It is written in mathematical language without which it is humanly impossible to comprehend a single word. This insistence on the importance of mathematics made Galileo the father of modern science. For this discovery is the very cornerstone of science. Today continued scientific progress depends on a more thorough a broader application of mathematical concepts. Thus we can discover more of the order the creator has built into the universe. Only thus can we read further in the great book of nature which lies open before us. [Music]