“…that vast book which stands forever opened 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.” — Galileo, the Father of Modern Science

Ali Kayaspor
Jun 25 · 13 min read

Everyone knows about honey bees. However, the bees have known what human mathematicians didn’t know for thousands of years. A honeybee may be the coolest creator in the universe. Its body is beautifully patterned, can fly wherever it wants, spends its time near beautiful flowers, produces the most delicious and incredible substance in nature, honey, and, most importantly, it is a great mathematician.The amount of knowledge they have of the world around them is comparable to graduating from the best schools of science and engineering. They show us that mathematics is the language of nature and science. Aristotle was one of the first to document the intriguing behavior of honey bees. For centuries afterwards, mathematicians have been become fascinated with bees.

I can announce with certainty that if Jon Snow was the King in the North, honey bees are kings in the insect kingdom. They have proven to be surprisingly intelligent compared to other creatures in the animal kingdom.

Tumblr, gif, The King in the North.

There is a famous misquote:

“If the bee disappears from the surface of the earth, man would have no more than four years to live.”

Humans could not understand the perfection of honey bees which were made by the Creator until the discovery of mathematics. But what makes honey bees nature’s greatest mathematicians? So far, we have 5 reasons to believe so:

  • They have the ability to produce a geometrically impressive waxy comb.
  • The reason why they prefer hexagons over other shapes
  • They can easily solve the “Travelling Salesman Problem”
  • They can grasp the idea of zero.
  • They can solve simple and basic mathematical questions.

A two week old honeybee basically becomes a wax printer. It can convert sugar from the honey into a waxy substance. At this point, an utterly impressive thing happens; a honeybee and its friends make arguably one of the most mathematically and architecturally efficient designs around a beehive, despite the fact that they have never studied theories of tessellations or engineering. But why does design matter to a bee? It is because of the material, wax.

Wax is extremely valuable to the bees and it is an expensive task to build a honeycomb. Thousands of honey bees travel thousands of miles, spend hundreds of thousands of hours to find nectar, transform it into the wax, and then fashion the wax to a precise pattern. All those steps make a honeycomb very valuable.

For the construction of a honeycomb, honey bees have to know the economy. They cannot make random shapes to make storage for the honey. There must be a geometrical shape which is more economical to build than any other. Here, “more economical” means honey bees need a shape which requires the least amount of wax to build and stores a lot of honey. In other words, the smallest lateral surface area and the maximum capacity. It is a mathematical approach.

Ideas.Ted.com, Honey Bees on a Honeycomb, Ideas.Ted.Com

For instance, a circle cell can be only economical if it stands alone. However, if you put circle cells together, they cannot be logical for the honeybee storage system. If you put circles together to make a tessellation or a pattern, you will notice gaps and wasted space in between them. A good economist doesn’t want any gaps. Furthermore, if they insist on making circular cells, since no walls can be shared, they have to make separate cell walls for every single cell, which is again a waste of wax material and a waste of time. That’s why honey bees need to work with regular polygons.

Here, there is something important I want to clarify. Some people are thinking that honey bees first used circular cells or other shapes of cells. After the evolution and trial-and-error, they finally got the best solution. I want to ask just one simple question here: Has there ever been a honeycomb fossil which is made of a different shape than a hexagon? No! For as long as they have been around, they have been using hexagonal cells.

A triangle, a square, a pentagon, a hexagon, and other blablagons are the options to choose to tessellate cells efficiently. For these shapes, one wall can do the work of two. It is a better way to save on material. If honey bees had not been smart, they could have gone with any one of these as the shape of their cells. Thank the divine force again, they went with the most magical and ideal structural shape, the hexagon. Modern science has proved this many times. By going with a hexagon, they can use the least amount of wax to build their cells, yet have the most amount of storage space to keep their honey afterward. I think we need to prove this claim now.

There is an interesting experiment to show why the hexagonal honeycomb is the best. Get six cups that surrounded by a belt. Pull the belt slowly and jostle the cups lightly. If you keep pulling, eventually you will get the honeycomb pattern with six cups arranged like those in a honeycomb- hexagonal.

Now, let’s look at why the hexagon is better than the other shapes mathematically. Here we need to answer these questions:

1- How should the volume be divided into shapes of equal size using the minimum amount of material?

2- What should the shape be?

These are deceivingly simple questions with very complicated answers. They are some of the oldest questions in history. Mathematicians call it “The Honeycomb Conjecture”. This article by Thomas C. Hales has many pages on the topic, explained with complex mathematics.

Let’s start with three possible arrangements. We’ll take an arrangement with triangles, then with hexagons, and then with squares.

Assume that our triangle, square and hexagon all have the same perimeter of 36.First, if we take a hexagon with the same perimeter as the triangle, the length of one side of the triangle will be 2 times bigger than the length of one side of the hexagon (3 ✕ 2 = 6). Here, we are able to fit four small triangles in our triangle; however, if we do the same thing with our hexagon, we notice that there is room to fix six triangles. So that shows the hexagon is better than the triangle.It’s a little more difficult to see why the squares are worse than the hexagons. But don’t worry! If you make a square with the same perimeter as a hexagon the hexagon will have a higher area than the square. Let’s say the length of one side of the hexagon is 6 cm. The perimeter will be 36 cm. The area of the hexagon will be 54 √3 cm² which is 93.530743… On the other hand, if we get a square with a perimeter 36 cm, the area of the square will be 81 cm². Thus the hexagon is more efficient. It is the same for all other shapes. The hexagon can easily beat them.
Self Drawn, Explanation of Why Hexagons are the Most Efficient Shape

You might think that since an octagon is rounder and has more sides than the hexagon, it will be better than the hexagon. However, octagons, like circle cells, do not fit together, so they cannot be the solution to the honeycomb. Only, a hexagon is made out of six sides that fit together to make a hexagonal honeycomb. Now we can confidently say that the hexagons are the best to use here.

Wikipedia.com, Euclidean Tilings by Convex Regular Polygons, Wikipedia.com

Philosophers and mathematicians believe that bees already know this mathematical approach when they are building the hexagonal honeycomb. Yes, it is possible, in the same way a football player might not need to understand physics before kicking a ball or a butterfly doesn’t have to understand the design of the camouflage of its own wings. So, honey bees don’t have to comprehend why the hexagonal honeycomb is best. But it is evident that a divine force is behind this remarkable natural event.

Of course, there is much more we can learn from a honeycomb. When I talked to a structural engineer on this topic, he showed me other advantages of the hexagon among all the other shapes. For instance, the hexagon is not just the most economical but also the strongest. Hexagonal cells make 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 3,000 times thinner than a sheet of paper. The understanding of the polariscope 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, only a computer can calculate them.


Basic math

The next reason why bees are great at math is because they are capable of understanding and solving basic problems. Although these insects have small brains, they can understand the mathematical concepts of addition and subtraction.

Scientists conducted an experiment with 14 free-flying bees and a Y-shaped maze. They published the results of the study in February 2019 edition of Scientific Advances.

At the entrance of the maze, they placed a set of visual stimuli: one to five shapes that were either blue or yellow; blue suggested addition while yellow meant subtraction. After seeing the initial number of shapes, the scientist encouraged bees to fly through a hole called the “chamber of decision” where they had to make a choice. They could either fly to the left or right of the maze. Both sides represented correct and incorrect answers. After noticing blue shapes at the entrance, if the bee chose the side with more shapes, then it was considered a correct answer and the little buzzing genius got a reward of sweet sugary water. If it chose the side where the number of shapes was fewer than those at the entrance, then it got punished by receiving bitter-tasting quinine for the incorrect answer. If the color of shapes at the beginning was yellow it meant that they had to fly next to a fewer number of them and the next chamber to get a reward. The correct answer changed randomly throughout the experiment to prevent the bees from learning the visit just one side of the maze. At first, bee made random choices. But they were later able to work out how to solve and learned that blue meant plus 1 and yellow was minus 1.

Then, to make it clear, the scientists set up a new experimental area. They changed the arithmetic problem to be solved and assigned randomly for each trial. They altered the side of the area where the correct answers were in order to eliminate the possibility that the bees might prefer one side over the other. They also eliminated the reward or punishment.

The lead author of RMIT study, Scarlet R. Howard, said, “What I hope people take away from this study is that insects are not unintelligent. They are smart and can do cognitively demanding things.” This experiment proves that bees have some mathematical ability. Of course, this is extremely simple task but it is quite impressive that an insect with a brain that measures about cubic millimeter can figure out basic math in 100 tries. The honeybee’s brain has about 1 million neurons which are nothing compared to a human’s hardware of 100 billion neurons. But if a brain about 20 thousand times smaller than ours can solve the equation indicates it could pay the way to a new approach in artificial intelligence (AI) and machine learning(ML) because the experiment shows us the math doesn’t require massive brain and some complex math can be done only by a limited number of nonhuman vertebrates. The power of honey bees’ tiny brains has implications for the future development of artificial intelligence, such as improving rapid learning.


Understanding an Abstract Concept

Zero is nothing but a number. It is one of the most important discoveries for humans. Zero has a very simple definition but still, a kid under the age of 6 struggles to comprehend the idea of zero because it is a very abstract and tricky. When we ask them to count, they start from one, not zero. If they have one toy, they say “I have one toy.” If they don’t have any toy, they don’t say “I have zero toys.” Zero is not even a number for them. However, a honeybee literally knows what is zero.

To understand that zero is less than one is a challenging thing because even a toddler is told zero represents something, such as the absence of a cookie, a toddler still doesn’t get that zero represents a quantity. For instance, in an experiment with four-year-old kids, researchers asked the kids to pick cards with the fewest dots, and when they compared a blank card and a card with one dot, less than half of the kids got the answer right. The same scientists from the University of RMIT were able to teach honeybees that zero is a quantity less than one.They presented bees with cards showing different numbers of dots. The bees were rewarded with sugar water when they selected the card with the smallest amount. After the bees were trained to always choose the lower number, the researchers upped the challenge and added blank cards to the test. Surprisingly, although the bees had never seen blank cards, they had chosen the blank card. This proved that bees can understand zero as a quantity on the number line because they chose the blank card when comparing it to a larger number like five or six than when they compared it to just one.

This is really big news for humans because if it only takes a bee-sized brain to get complex abstract concepts, maybe there are far more efficient ways to design AI. We could make better, more efficient computers one day.


The Complex Traveling Salesman Problem and Waggle Dance

Finally and most importantly why honey bees are great at math is because they can easily solve the complex Traveling Salesman problem that can keep computers busy for days. In other words, they can find the optimum route to minimize their cost. To solve the Salesman problem, computers go about finding the shortest route by looking at every possible route, compare their lengths, and choosing the shortest route. However, scientists have just discovered that foraging honey bees can find the shortest route between flowers and come to the same conclusion as a computer. They don’t even need to calculate all of the possibilities to find the shortest route. This sort of active memory and learning is what we previously thought was something only larger-brained animals were capable of but the bee proved us wrong.

This is not the end of the story. There is something more interesting about this problem. After the honey bees who find the food source go back to their hive, different honey bees go to the food source individually, not as a group. Scientists realized that honey bees communicate amongst themselves in a very interesting way.

A foraging bee returns to the hive and waggle about excitedly in a figure-eight pattern before sharing the collected pollen and nectar with their hive mates. The bee’s dance gives the rest of the colony a lot of information. If another bee comes from the same food source the angle of rotation precisely matches the angle between the feeding stations and the hive. On the other hand, if a returning bee from a different feeding source danced differently from bees that arrived from the other location.

Wikipedia, Waggle Dance, Wikipedia.com

Honey bees’ eyes can detect ultraviolet and polarized light from the Sun and they use their eyes to use as the best solar compass determine the precise location of the Sun at any time. Also, bees have an internal clock that helps bees estimate the new position of the Sun as it travels across the sky.

So honey bees make special angles by using their solar compass and internal clock while they are dancing. This lets fellow workers always what angle to travel away from the Sun to find food.

Their special dance also contains information about the distance to a food source. The longer time spent in this part of the dance means that the food is further away. Shorter durations mean that the food is closer by. Then if she’s really excited about it like really going for it you know that it’s a proper good source of food.

Humans have much to learn from bees who accomplished this extraordinary feat with a brain the size of grass seed.


So those are the reasons why mathematicians think bees make one of nature’s greatest mathematicians. Mathematics is the universal language and when you look at a perfect honeycomb you see a shadow of that language. Mathematics made real by bees and I certainly believe that all the instilled knowledge in honey bees is coming from the divine force. 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.

I wish honey bees were my math teacher or my friends so they could do my homework.

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.

Ali Kayaspor

Written by

Math Teacher. Mathematics. Soccer player. That’s why it is Maradona not Madonna. I had a mustache like a comrade when I was 2. — alikayaspor@gmail.com —

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.

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