Bees are one of the best known pollinators in the botanical world. The sweetness of Nature relies on the art of symmetry of the relationship between flower and the Bee. The result is the collection of sweet, luscious, nectar. Once all the nectar has been dutifully collected, the bee comes back to the hive, where the nectar will turn into nourishment for her and the rest of the colony. What is fascinating is shape of the hive she is in, the storage bins for the nectar. Bees are also one of the best geometers. So why are the honey comb cells hexagonal? Well, it is time to be a kid and blow some bubbles first.
The bubble, or the sphere, is the most efficient shape to hold what is inside, a given volume. It is the perfect relationship of surface area to volume. Individually, the spheres can exist in maintaining this ratio without distortion. However, if you go get a glass of milk and a reusable straw, you will find that competition for a given area will change the geometry of this perfect ratio. In other words, the geometry of the bubble will change.
The story of milk and bubbles actually began, for me, about five years ago when writing a math blog about lessons to do at home with children. There is nothing more fun than to get a glass of milk, a straw, and blow bubbles. While you are having some fun, you could also notice how the spherical bubbles would get bigger and bigger and then quickly form other little bubbles to fill the gaps, in between the bigger ones. Then, as more and more seemed to be created, those wonderful spherically shaped bubbles would then form into something hexagonal in shape (link to "Math on the Mountain: The Geometry of Streams, Bubbles, and Chocolate Milk").
The same happens to a moving stream of water, along the desert wash. As more and more bubbles form, they are doing their best in aerial/water tiling, as they press against the other. Tiling happens in Nature when shapes are trying to fit into the most efficient pattern that will result in having less spaces/gaps in between the next shape. Sometimes that results in little tiny bubbles fitting into the spaces, and sometimes those bubbles will take on a hexagonal form to them, just like what happens to the bee hive "storage bins" for their nectar.
In tiling, one of the shapes that can be used individually over and over again, are hexagons, not spheres. Over thousands of years and many generations, Bees have used this six-sided polygon, the next best thing to spheres and maintaining that ratio, to be the most efficient way to store their hard earned harvested nectar, to provide a food source for the rest of their colony. Symmetry, geometry, finding that perfect ratio, it is all there. It is there in Nature.
Cornell University (May 2004). Bubbles! http://pi.math.cornell.edu/~mec/2003-2004/geometry/bubbles/bubbles.html
Grossman, C. (2016). Math on the Mountain: The Geometry of Streams, Bubbles, and Chocolate Milk. https://learningmathwithmom.wordpress.com/2016/09/01/math-on-the-mountain-the-geometry-of-streams-bubbles-and-chocolate-milk/
Illustrative Mathematics (2016). Hexagonal Pattern of Beehives. https://tasks.illustrativemathematics.org/content-standards/tasks/1126
I am not the first, nor the last of expressing and sharing the beauty of mathematics in Nature. What I will share in this blog are thoughts, experiences, and lessons learned to validate life, both human and botanical, living mathematically.