Hexagonal Patterns

Stuart Ball Betty Taylor Brenda Weddell

Class: Geometry

Materials: Hexagonal worksheets (included with handout), colored pencils, random number generator for integers between 1 and 6 (e.g., 1 or 2 dice).

Goals: Develop an appreciation for geometric patterns in nature, combine random events with geometric topics, recognize patterns.

Time Required: 1 day

Background: None required, although a quick discussion of randomness and the use of random numbers might be necessary.

Technology Required: None

Problem:

Bees always build honeycombs in a flat hexagonal pattern. Each cell in the honeycomb is actually a 'nursery' for a young bee. Worker bees place a grub (infant bee) in each cell along with plenty of food, after which the top of the cells is sealed with beeswax. If it is undisturbed for a few days, a fully-grown bee will eventually eat its way through the wax covering and immediately go to work in the hive.

Bees have a tiny, though deadly, enemy that attacks them during the vulnerable stage when the grub is growing inside a honeycomb cell. Worker bees that are depositing the grubs and food sometimes carry tiny mites on the fur of their bodies. These mites will often fall into the cells as they are being filled. For the sake of this investigation, let's assume that each mite deposits two larvae within the sealed cell, after which the parent mite dies. These two larvae eat the food that was intended for the bee grub, leaving it to starve to death. Eventually, the two mites mature and eat their way out of the cell. Rather than eat through the hard top seal, they both prefer to eat through the softer honeycomb walls into adjoining cells. If there is food in the cell that a mite chooses, it deposits two new larvae and the process repeats. (This type of infestation often wipes out entire hives very quickly.)

To represent this situation, we will use a hexagonal grid to represent the honeycomb -- see back page. Imagine the first mite deposits its two larvae in cell X. These two larvae mature into adult mites, which move into two (randomly-selected) adjoining cells. (We'll assume the two mites never select the same cell.) If a mite selects a cell which contains food, it will survive and lay two new larvae. If it selects a previously-used cell, the mite dies and does not deposit its larvae.

Starting with cell X, you are to simulate the destructive effect of 10 generations of mites.

1. How will you randomly select the 2 adjoining cells that the larvae will eat into in each generation?

2. What assumptions are you going to make as part of your model? Which were given in the statement of the problem, and which are your own?

3. Since you are representing random events, no two experiments will be the same. How will you look for patterns that occur despite this randomness?

It is not clear what effect the starting location has on the eventual pattern of destruction. To investigate this aspect, you will notice that another cell toward the edge of the honeycomb has been marked Y. Repeat your simulation using this cell as the starting point. Again, you might want to repeat it three or four times to try to remove some of the effects of the randomness.

4. Does the starting point affect the destruction to the hive? If so, in what way? Can you think of any explanations?

Extensions:

What assumptions were made in your investigation that you found unreasonable? For each assumption that you find objectionable, how could you change the simulation to better imitate the real world? (Warning: Just select two assumptions that you find objectionable and deal with them. Otherwise, your model could get very, very complicated and that defeats the purpose behind this sort of exercise.)

Teacher Notes:

Answers to questions posed in exercise:

1. To select the 2 cells that will be invaded by mites, the students should suggest some method that ensures a pair of random whole numbers between 1 and 6. For example, a pair of dice could be rolled. Assuming the surrounding cells are numbered 1 through 6 (possibly in clockwise order from the top), the dice will give the numbers of the 2 cells that will be taken over by the next generation. If the same number appears on both dice, one could be re-rolled until a different number appears. Students could also use numbered slips of paper drawn blindly from a container, a random number table, a computer or calculator that gives random numbers, and so on.

2. Some of the assumptions that students may cite are listed here -- the list is by no means complete. Those marked by an asterisk (*) are taken from the statement of the problem; those marked by a checkmark () would be student-generated.

* The bees always build hexagonally-shaped honeycombs.

 The hexagons in the honeycomb are all congruent (same size and shape).

* Only one bee grub is placed in each cell.

* Only one mite is deposited in the starting cell (X or Y) by a worker bee.

 The only mite that is initially deposited is the one in the starting cell.

 After one has been placed in the starting cell, no other mites are introduced into the

cell (by worker bees or any other means).

* Each mite produces exactly 2 larvae.

 The larvae need no more food than the worker bees deposit in the cell.

* Both larvae survive to adulthood.

* The larvae leave the cell through the walls and not through the top or bottom.

* The 2 young mites eat their way into different cells.

* If a mite eats into a cell without food, it dies; otherwise, it lays 2 new larvae.

 Cells are only used once and then remain empty.

 If a mite eats its way through an outer wall of the honeycomb, it dies.

 The mites have no predators.

3. To overcome the randomness of the simulation, each group should repeat each part of the experiment at least three or four times. They can do this by starting on a fresh hexagonal grid each time, or by using colored pencils. Then each individual or group can share their results with the class. A few examples of each simulation have been provided on the next few pages.

4. Generally, an infestation that starts in the center (X) results in more widespread destruction than one that starts near an edge.

Evaluation considerations: The student responses should be judged on their merit. There are no completely right or wrong answers. Does the response mention valid assumptions, and are their suggestions for improving the simulation reasonable ways to account for these assumptions?

Funded in part by the National Science Foundation and Indiana University 1995