Honeybees and yellow jackets do not resemble mathematicians; for one thing, they are much smaller. However, the insects may solve a common architectural challenge collectively by employing a geometric solution that they evolved independently of one another.
These bees and wasps eventually need to increase the size of the hexagonal cells that make up their nests as their colonies grow. However, nest material is expensive, and combining hexagons of varying sizes into a single continuous array is difficult. Honeybees and wasps have both solved this challenge by combining some pairs of five-sided and seven-sided cells, which bridge the gap between different sizes of six-sided hexagons.
Female workers rear the children of their mother, the queen, in social insect colonies such as honeybees and certain wasps. They do this in hexagonal cells constructed by honeybees from wax and wasps from paper.
At some point in its life cycle, the colony must transition from raising workers to raising reproductives like males and new queens. These reproductives are frequently larger than the workers, implying that the hexagonal cells must also grow in size. If you have two different sizes of hexagons and want to group the small ones on one side and the huge ones on the other, you're going to have some sort of problem fitting them together.
To figure out how bees and wasps solve this tiling puzzle, analysed 115 images of colonies of five species of honeybee (Apis mellifera, A. cerana, A. dorsata, A. florea and A. andreniformis), four species of Vespula wasp (V. vulgaris, V. maculifrons, V. flavopilosa and V. shidai), commonly known in North America as yellow jackets, and one species of paper wasp (Metapolybia mesoamerica).
The scientists gathered data from 22,745 cells using an automatic image analysis tool built by team member Kirstin Petersen, a roboticist at Cornell University, such as cell wall lengths and how many neighbours each cell has. The automated approach enabled the team to collect data from irregular cells that aren't ideal hexagons, which many scientists had overlooked due to the difficulties of physically measuring them. These ostensibly misshapen cells proved to be anything but.
To overcome the gap between small worker cells and large reproductive cells, all bees and wasps built pairs of contiguous five-sided and seven-sided cells. Because a five-seven pairing has the same number of open sides as a hexagonal pair — both types of conjoined pairings have 10 sides accessible to connect to other cells — it does not disturb the pattern. The greater size of the seven-sided cell permits the bees and wasps to start constructing larger hexagons on the other side of it almost immediately.
A Cornell University computer scientist created a mathematical model of this method and discovered that what the bees and wasps are doing is close to the best geometric solution. A Delaunay triangulation is the most efficient approach to construct an array of forms such that each cell is large enough to rear a baby bee or wasp.
According to Napp's concept, adding larger hexagons to the nest gradually moves the entire array away from perfection, causing gaps to occur or workers to construct an unusable cell to keep the nest together. Across all bee and wasp species, around 85 percent of all non hexagonal cells are in five-seven pairs, as predicted by the model.
The bees and wasps utilised in this study evolved over 179 million years and use different materials to build their homes. “However, both evolved to apply this five-seven method for transitioning between hexagon sizes, demonstrating that "evolution has a tendency to solve challenges optimally."