Mathematical aspects of bees

Summary:Geometry explains why honeycombs are made of hexagonal cells, while bee movement patterns communicate information visually.

Honeycombs are remarkable for their beauty, precision, and symmetry. The honeycomb corresponds to a mathematical concept known as a “tiling of the plane.” That bees use regular hexagons for this tiling (built to a remarkable level of precision) has fascinated human beings throughout history. At the end of the twentieth century, mathematician Thomas Hales rigorously proved a long-standing conjecture that fully justifies to humans what the bees have apparently known all along: the most efficient way to repeatedly enclose a fixed amount of storage space is to use regular hexagons to form the boundaries.

Honeycomb: How to Choose a Cell

Bees use honeycomb cells for storage. It takes work and material (wax) to create the boundary of each cell, so the bees want cells with as little boundary (perimeter) as possible, given that each cell should enclose a certain amount of storage (area). If a bee only needed to make one cell to store honey, it would likely use a shape other than a regular hexagon. For instance, a regular octagon holding the same area has less perimeter; a regular decagon will have less perimeter still. The more sides a polygon has, the smaller the perimeter will be, with the circle having the smallest perimeter-to-area ratio. That a circle is the least-perimeter shape to enclose a given area is a famous problem that goes back to the wonderful tale of Queen Dido of Tyre.

For example, suppose a bee wanted to enclose one square unit of area. The square that accomplishes this has a perimeter of 4. If the bee used an equilateral triangle instead, the necessary perimeter is larger, about 4.56. But the regular hexagon’s perimeter is smaller, at just over 3.72. The pattern of increasing the number of sides leading to a lower perimeter holds for all whole numbers n > 2, and every such regular polygon enclosing one unit of area has greater perimeter than a circle holding the same area. The circle that encloses one unit of area has a perimeter of approximately 3.54.

Tilings: Fitting the Cells Together

Bees do not need just one cell; they need many consecutive cells in which to place their honey, and therefore essentially have to create a “tiling” (a pattern involving polygons that will completely cover their work space without overlapping while leaving no space unused). Circular cells simply don’t fit together as well because there are gaps between consecutive circles.

Many different kinds of floors and ceilings are tiled—usually with congruent squares or rectangles. Why don’t bees use square cells in their honeycomb, rather than hexagons? Or equilateral triangles? It turns out that equilateral triangles, squares, and regular hexagons can all be used to tile the plane, as shown in the figures below. Bees choose hexagons from among these three options since a regular hexagon of unit area uses less perimeter (wax) than does a square or equilateral triangle; the hexagon is a more efficient choice (see Figure 2).

So why not use regular octagons? Here it is not the efficiency of the individual cell that governs the choice but rather the overall packing of them: regular octagons cannot be used to tile the plane.

To understand why triangles, squares, and hexagons tile the plane, but octagons do not, observe that in a regular polygon with n sides, the sum of its interior angles is 180(n-2) degrees, and each of its n individual interior angles has the measure

For instance, with the square, each interior angle has the measure

Four squares arranged at a single vertex fit together perfectly, creating a full 360 degrees around the shared corner. Likewise, six equilateral triangles (each having 60-degree angles) can fit together perfectly for a full 360 degrees, as can three regular hexagons with their 120-degree interior angles.

But for the octagon with n=8, each interior angle has the measure of 135 degrees. Three octagons put together at a shared vertex would have 135×3=405 degrees, which is simply impossible—as would be attempting to only have two octagons meet at a single vertex. Regardless of the number of sides of the regular polygon, the measure of the polygon’s interior angle will need to divide evenly into 360 degrees. This forces

to be an integer, and the only values of n for which that is true are n=3, 4, and 6: triangle, square, and hexagon! That the only ways to tile a flat surface using congruent regular polygons are with triangles, squares, or hexagons is a result often taught in high school geometry courses.

Irregular and Non-Polygonal Tilings

Since the time of the ancient Greeks, mathematicians conjectured that among all the ways to tile the plane so that each tile encloses just one unit of area, the way that uses the least perimeter is the tiling that uses all regular hexagons. This conjecture is much harder than it sounds to prove: one must consider irregular polygons (with sides of different lengths), as well as the possibility that the sides of some tiles might be curved. The first possibility is not too difficult to eliminate. For instance, it is straightforward to show that a regular hexagon with all sides of equal length will use less perimeter than any other hexagon to enclose the same area.

But the second possibility—using non-polygonal shapes—proved to be much, much more challenging. In this situation, one must consider the possibility of a shape that bows out on one side and, to fit into a tiling, bows in on another. Obviously, the part that bows out picks up area, while the part that bows in loses area. In 1999, mathematician Thomas Hales proved that any advantage that comes from a side of the tile bowing out is more than cancelled out by the disadvantage that follows from another side having to bow in. Thus, the ideal tile is one that has no bulges: a polygon!

What Professor Hales proved is essentially what the bees knew all along: of all possible tilings, the one using regular hexagons is the most efficient way to enclose cells of the same area.

Other Mathematical Aspects of Bees

Another way that mathematics relates to bees is when mathematicians work with bee researchers to solve problems such as those related to viral disease infection and pollination. Mathematics is also used to model the ways in which bees communicate locations. When a bee finds a source of food, it returns to the hive and performs an elaborate dance that conveys the direction and distance from the hive. Ethologist Karl von Frisch was one of the first to explore the meaning of the honeybee dance, and he won a Nobel Prize for his work. The angle that the bee dances expresses the direction. For example, if a bee dances in a straight line toward the upper part of the hive, then the flowers are located in the direction of the sun. The bee also takes into account the fact that the sun moves; the angle it describes inside the hive changes as the sun does. The duration of the dance and the number of vibrations give the exact distance. Other features of the dance remained unexplained until Barbara Shipman theorized that the honeybee’s complex choreography is a projection of a six-dimensional space, and she was able to use this representation to reproduce the entire bee dance in all its parts and variations. To her, this implies that bees can sense the quantum world, although some researchers dispute her conclusions.

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