Real Mathematics – Geometry #13

Baking Cookies

You started craving for some chocolate-chipped cookies and you are trying to convince your mother to bake some… or maybe more than some. Eventually you and her met halfway: Your mom will cook only one oven tray of cookies and your task is to place them on the tray. Obviously all cookies must have identical shapes. They can’t be placed randomly and they should not overlap on the tray. In the end you have to come up with a system that will give you the most cookies possible.

 

Dimensions of the oven tray: 46,5 cm x 37,5 cm.

Diameter of a single cookie: 5 cm.

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Q: How many cookies can you fit on this oven tray?

Honeycomb

First we must analyze how honey is produced in order to solve the cookie problem. Honey bees use fantastic mathematics and their masterpiece architecture is honeycomb.

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Have you ever thought why do honey bees construct honeycomb as they are (which is in a hexagonal shape)?

Honey bees are in need of storage units for the honey they produce. For now try to imagine those storage units in two-dimensional plane. This way honey bee problem turns into a problem of finding the optimal regular geometrical shape in order to tessellate a paper.

Responsibility of the Honey Bee: “I must tessellate a paper with a regular geometrical shape and I shall do that with creating the maximum area with minimum circumference.”

(Maximum area: More honey storage. Minimum circumference: Less time spent on constructing the storage.)

Three Regular Polygons

There are only three regular geometrical shapes to fill a paper without a gap: Equilateral triangle, square and hexagon.

bad tessellates
Here is how 7, 8 and 9-sided regular polygons look like on a paper. You could try a polygon and see the result on your own.

Let’s start with a square that has side length of 2 units. It means the area of one square is 4 units and its circumference is 8 units.

 

Now I’ll consider constructing an equilateral triangle that has area of 4 units. Its circumference would make a little over 9 units. This means that equilateral triangle tessellations will take more time for the honey bee.

 

Next one is the hexagon. Whenever a hexagon has an area of 4 units, its circumference will be around 7,45 units. This result shows that a hexagon requires less time to be built than a square does. This is why our honey bee chooses hexagonal tessellations in order to construct its honeycomb.

 

Circle to Hexagon

We still don’t know how honeycombs are being constructed. Scientists consider two possible scenarios for it: Bees use either circles or hexagons in the construction of honeycombs. One theory suggests that bees build circle storages and those storages turn into hexagonal shapes within time.

Following picture illustrates how circles and hexagons are related with one another:

 

One wonders…

Solution of the cookie problem is hidden inside the relationship between circle and hexagon. Then please find how many cookies you can place on the oven tray.

M. Serkan Kalaycıoğlu

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